Patterning technology for folded sheet structures

ABSTRACT

The present invention supplies practical procedures, functions or techniques for folding tessellations. Several tessellation crease pattern techniques, and the three-dimensional folded configuration are given. Additionally several new forming processes, including mathematical methods for describing the material flow are discloseddoubly-periodic folding of materials that name the doubly-periodic folded (DPF) surface, including vertices, edges, and facets, at any stage of the folding. This information is necessary for designing tooling and forming equipment, for analyzing strength and deflections of the DPFs under a variety of conditions, for modeling the physical properties of DPF laminations and composite structures, for understanding the acoustic or other wave absorption/diffusion/reflection characteristics, and for analyzing and optimizing the structure of DPFs in any other physical situation. Fundamental methods and procedures for designing and generating DPF materials include ways for defining the tessellation crease patterns, the folding process, and the three-dimensional folded configuration. The ways are mathematically sound in that they can be extended to a theorem/proof format.

This application is based on and claims priority from provisionalapplication 60/232,416 filed Sep. 14, 2000.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to folded tessellations and other foldedstrucures. More particularly, the present invention relates to specificconfigurations, and patterning methods, applicable to the unfoldedsheet, the three-dimensional folded structure, proccesses oftransforming a sheet to a folded strucure, and machine descriptions forthe same.

2. Description of the Related Art

A well-known problem in the art of designing and forming materials intofolded networks is that with the exception of deformation at the fold,the material is not signifgantly streched, and this imposes simultaneousconstraints on the vertices, edges, and facets of a proposed structureand on the process of forming such a structure. However foldedstructures have many advantages over structures produced by other meanssuch as casting, stamping or assembling, such as cost of manufacture andthe versatility to many sheet materials.

SUMMARY OF THE INVENTION

Accordingly, the present invention discloses our methods and proceduresfor designing and generating folding networks and tessellations. Severalmeans for defining the crease patterns, and the three-dimensional foldedconfiguration are given. Additionally several new forming processes,including mathematical methods for describing the material flow aredisclosed. The description and use of practical methods, datastructures, functions and techniques that name the folded surface,including vertices, edges, and facets, and that describe the formingprocesses for sheet material is disclosed. This information is valuablefor designing tooling and forming equipment, for analyzing strength anddeflections of the DPFs under a variety of conditions, for modeling thephysical properties of DPF laminations and composite structures, forunderstanding the acoustic or other wave absorption/diffusion/reflectioncharacteristics, and for analyzing and optimizing the structure of DPFsin any other physical situation. Many other aspects of our foldingtechnology are also presented.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates the creasing along a tessellation to produce a PDF.

FIG. 2 illustrates the column and row phenomenon.

FIG. 3 is a chart comparing various folding methods according to thepresent invention.

FIG. 4 illustrates arrays of DPFs generated by using Row Cross Section(RCS) and Colum Cross Section (CCS).

FIG. 5 illustrates Z-values of RCS are the offsets on the CCS.

FIG. 6 shows a simple folding tessellation according to the presentinvention.

FIG. 7 illustrates steps (a) through (d) in the Wave-TessellationMethod.

FIG. 8 illustrates fold convexity at a Vertex.

FIG. 9 illustrates fold locations on a strip.

FIG. 10 illustrates the defining of a strip map by polygoncorrespondence.

FIG. 11 illustrates several strip maps according to the presentinvention.

FIG. 12 illustrates the extending of a strip map to construct DPFs.

FIG. 13 shows vertex calculations with strip-maps to construct DPFs.

FIG. 14 is a flowchart of the procedure for determining thecorrespondence function from an Unfolded Sheet to a Folded Sheetaccording to an embodiment of the present invention.

FIG. 15 illustrates compositions of Local Isometries.

FIG. 16 illustrates the geometrical comparison of DPF-VertexCalculations.

FIG. 17 illustrates the trigonometric relationships between entry data.

FIG. 18 illustrates two reflection patterns for a line.

FIG. 19 illustrates a local Isometry applied to two surfaces.

FIG. 20 illustrates that almost all the components cannot foldgradually.

FIG. 21 illustrates the conventional prior art creasing process.

FIG. 22 illustrates the novel creasing process according to the presentinvention.

FIG. 23 shows a comparison of the conventional creasing process versusthe novel creasing process according to the present invention.

FIG. 24 illustrates the implementation of a two phase forming processusing rollers to fold.

FIG. 25 illustrates parameterized RED and CCS data for a Chevron Patternapplication.

FIG. 26 illustrates vertex coordinates with DPF pattern type parameters.

FIG. 27 provides a table illustrating coordinates for the vertex of achevron pattern using the wave-fold method and parameterized RED and CCSaccording to the present invention.

FIG. 28 illustrates a plurality of wave patterns w1-19 according to thepresent invention.

FIG. 29 illustrates another plurality of wave patterns w10-w17 accordingto the present invention.

FIG. 30 illustrates yet another plurality of wave patterns w18-w26according to the present invention.

FIG. 31 illustrates a machine implementing a process according to thepresent invention.

FIG. 32 illustrates five different frames with parameters overhead.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(Introduction and Definitions)

It should be understood by persons of ordinary skill in the art that themethods and procedures described here in Part I refer to folding anidealized infinitely thin plane. The plane is deformed, without changingany intrinsic lengths along the surface itself, to produce multi-facetedgeometric structures. Any surface folded from a plane will havezero-curvature. This means the Jacobian of the Gauss map is zero onsmooth regions, that the sum of the geodesic curvatures along an edgesingularity (crease), when measured from the two sides of the edge, willtotal zero, and that at each vertex the cone angle will total 2π=360degrees. The zero-curvature results from the fact that the material isnot stretched by folding, and so the surface remains isometric to theplane.

In practice a sheet has finite thickness and may have intentional orunintentional distortions in superposition to the ideal processdescribed here. The idealized zero-curvature surface is thearchitectural base, from which close to zero-curvature surfaces will bedesigned. For instance, even if a folding pattern for sheet material isdesigned identically to the folding pattern for a plane, the physicalsurface will not have zero-curvature perfectly, because the thickness ofthe sheet causes a bend radius at each fold, and generally this forcesvery small regions of positive curvature and negative curvature near thevertices. Moreover it may be desirable to select an idealized foldingpattern to plan the major displacement of the sheet material, and thenexploit the plasticity of the material to make minor adjustments to thesurface configuration. The design difficulties and manufacturingadvantages for these surfaces close to zero-curvature surfaces are verymuch the same as for zero-curvature surfaces, and so this is a valuableapplication of our technology. Typically close to zero-curvaturesurfaces and zero-curvature surfaces differ by a minor perturbation.Thus by using the idealized zero-curvature surface as an architecturalbase, these valuable close to zero-curvature surfaces are boththeoretically and in practice an application of the our technology.However the key step is determining the idealized architectural base,and this is the focus of our numerical and geometrical methods.

The technology disclosed here also applies to repetitive foldingpatterns. These idealized surfaces may have one or two directions ofrepetition. The repetitions may correspond to translation or rotationalsymmetry. A doubly-periodic surface with zero curvature is called afolded tessellation. In practice, folded tessellations are finite andmay have intentional or unintentional imperfections and so may have noexact translational or rotational symmetries.

In this disclosure the term DPF refers to the patterned surfacesgenerated by our methods and technology presented here. The idealizedDPF will have exactly zero curvature. Moreover, by entering non-periodicdata into the procedures and methodology, many valuable non-periodicfolding patterns can be designed by DPF technology. By entering periodicdata into our methodologies, the resulting surfaces will be foldedtessellations. A physical folded sheet is called a DPF if its design hasan architectural base from an idealized DPF, although the scope of DPFtechnology is much broader, including sheet material with regions orcomponents that are designed in part by DPF procedures and methodology,with or without intentional or unintentional variations into close tozero curvature geometry.

(In reference to other documents, the meaning of the accronym DPF hasgone through a series of changes, and at one time refered abstractly todoubly-periodic flat surfaces that may or may not have existed in onlythree-dimensions.)

Each of our various designing methods has its own generalizations fordefining DPFs. The main class of DPFs are the doubly-periodic foldingstructures describable by all of our designing methods. The simplestclass of DPFs is included in the main class and has a more obvious arraystructure and also the straightforward two translation symmetries, andwill be described first. Until otherwise stated, the discussion in partI will refer to DPFs in the simplest class. The terms and techniquesgiven are part of the invention, and generally will not apply to foldingstructures produced outside our technology.

FIG. 1 shows four stages in the Uniform DPF Process for producing atypical DPF in the simplest class. In the Uniform DPF Process, theindividual facets are assumed to be rigid and only the fold angles ofthe edges may change. In our Uniform process the DPF flexes with oneparameter of motion. Data from any one stage will determine the entireprocess. In FIG. 1A-D this parameter, called the flex parameter, variesfrom zero to its maximum π/2=90 Degrees.

Notice the edges form a net (also called a mesh or a graph) on thesurface, with degree four vertices and quadrilateral regions. By analogywith the usual rectangular grid, this enables the edges to be dividedinto two types, according to whether they advance in the row or columndirection. In one direction (for the simplest class) the edges formparallel lines in the fold crease tessellation, as shown in FIG. 1A. Theend-to-end chains in this direction are considered to be the columns ofedges (left to right in FIGS. 1& 2). The other edges form end-to-endchains that are considered to be the rows of edges (top to bottom in thefigures). In FIG. 2, A an exemplary row and column are shown in boldfacelines.

It can be seen that at any stage of the folding process, the edges inany given row all have the same fold convexity and are coplanar, andthat the edges in any given column alternate sequentially in foldconvexity and are coplanar in a vertically oriented plane.

Two other observations will be necessary concerning cross sections inthe column and row directions of a folded DPF. Of course, the facetsconnect in the row and column direction to form chains called a row andcolumn offacets. A row of facets will be bounded on either side by a rowof edges, and will have its facets connected successively acrossindividual column edges. At any stage of folding all of the column edgeswithin a row of facets are parallel. Furthermore, an entire row offacets can be developed into a family of line segments parallel to itscolumn edges, as shown in FIG. 2C. Viewing a row of facets as a familyof parallel line segments, and extending the line segments if necessaryto a larger ruled surface, one chooses a plane normal to the parallellines of the ruled surface and defines the row cross-section to be theintersection of the plane with the ruled surface. Each row of facetswill be seen to have the same row cross section—although if one uses theorientations available the row cross section alternates in mirror imagewith each subsequent row of facets.

In the fold crease tessellation of an unfolded DPF, the two columns ofedges that bound a column of facets on both sides form parallel lines.During the Uniform DPF Process, the vertical planes that contain thecolumns of edges are all parallel. Moreover note any other line drawn onthe tessellation parallel to the column edges will also lie in avertical plane throughout the process. The plane will be in factparallel to the planes containing the columns of edges. This foldeddrawn line is called a column cross-section. Several columncross-sections are drawn on one column of edges in the DPF in FIG. 2D.

It is sufficient (for the simplest class still) to specify theequivalent of one row and one column and their intersection point todetermine the entire DPF surface. Our procedures and methods differ inhow the row and column data is implied mathematically.

As shown in FIG. 3, for the row data, three presentations are currentlyavailable:

-   1. A row of edges in the tessellation (RET).-   2. The row cross-section (RCS).-   3. A row of edges on the folded DPF (RED).

As these are piecewise-linear coplanar curves, it suffices to enter thevertices on a plane for one unit of repetition with the number ofrepetitions desired. This is the prefered data entry in our computerprograms, although other methods, such as describing the edges asincremental vectors in polar coordinates, have also been used.

In addition, as shown in FIG. 3, for the column data, threepresentations are currently available:

-   1. A column of edges in the tessellation, augmented to give the    relative amplitudes and spacing of successive rows of the    tessellation (CET).-   2. The column cross-section (CCS).-   3. A column strip map (CSM).

For the first two it again suffices to enter the two-dimensionalvertices for one unit of repetition and the number of repetitionsdesired. A similar reduction is possible for the strip map and isdiscussed, infra, with the Strip-Map Method in with regard to strip mapdesign and shown in FIGS. 9-13.

FIG. 3 summarizes our procedures and methods by their row/column datarequirements, and gives an overview of their respective advantages.

As coordinates become necessary x, y and z will denote the row, column,and vertical directions respectively. The row data RCS and RED will besupplied on an XZ plane, RET on an XY plane, and the column datasupplied on an YZ plane. The RCS and RED Z coordinates are rotatedrelative to the three dimensional z coordinate.

With exception of tessellation data, the row and column data are usuallyassumed to refer to the to the DPF in its intended position. For theWave-Tessellation Method, the flex parameter must be given additionallyin some form. Each method also has the option for changing the intendedflex parameter if desired, and can implement the change within theMethod by adjusting the data entered to fit the new stage of folding asdescribed, infra.

For the main class of DPFs it is also possible to translate betweenmethods, as described in Section 9. Thus the designer may visualize theDPF from several perspectives and create a DPF to meet variously statedrequirements. Moreover, while editing, rescaling, and partially foldingor unfolding, the user may jump back and forth between methods. Thisflexibility enables the data to be developed in the terms of the methodsmost convenient for the application, so one can find and optimize thechoice of DPF readily.

The methods share some common features relating to the format of thedata entered and the data out-putted. To generate folded tessellations,first for efficiency one may enter just one unit of the row and columndata along with their respective repetition numbers. One may apply therepetitions to the entry data to generate the full row and columninformation and then apply the procedures to calculate the DPF surface,or one may input into the procedure one row unit and one column unit,apply the procedure, and then apply the repetitions in the row andcolumn direction to generated the full surface. In some cases, such aswhen the row and column data do not have a standard repetition, onechooses to use the full row and column data for the entry data.

Also the piecewise linear (PL) structure of the entry data and theout-putted DPF surface means that they may be linearly interpolated fromtheir vertices, and the interpolation can occur before or after theapplication of the procedures. For instance with the Two Cross-SectionMethod, for each pair of segments, with one from RCS and one from CCS,the method will generate a quadrilateral facet on the DPF. The methodtakes the two endpoints of the two segments to the four vertices of thequadrilateral. However, the convex-hull of the four output verticesdefines the quadrilateral. Thus it may be computationally more efficientto use just the vertices of the PL curves to generate the vertices ofthe facets of the DPF and then interpolate to construct the facetregions, then to use the fully connected PL curves to generate the facetregions. The methodology to exploit this fully for these folding maps isdescribed in Section 7.

Each of the computer codes for the methods also have rescalingsubroutines and other convenient techniques for adjusting the elementsin the data entered. For instance often one wishes to experiment withina DPF pattern type. This implies there is a type for both the row dataand column data, defined by restricting certain of their symmetries andother characteristics. Within the restrictions, one may parameterize thepossibilities allowed for the row and column data types and use this asa front end for our computational procedures. The user then experimentsefficiently with these parameters, which generate the row and columndata, which in turn generates the DPF within the selected pattern type.This is further discussed in Section 11.

3 Two Cross Section Method

3.1 Overview

This method generates DPFs directly in three-space and has variousgeneralizations. In the simplest form, the user enters a row and columncross sections by simply naming their vertices on a coordinate plane.The cross-sections can be designed effortlessly. The resulting infinitetable contains DPFs with great commercial application. Moreover thedirect relationship between the data entered and the resulting structurewill enable engineers to design materials with custom performanceproperties. For both chains a variety of front ends are possible, suchas entering a single unit of repetition may be entered along with thenumber of repetitions desired in each direction, or using a computercode to generate the chains within selected parametric conditions. Thedesigner typically selects a general DPF pattern type from the infinitearray for its structural characteristics, and then optimizes theparameters in that pattern type by adjusting the parameters in thecorresponding RCS and CCS wave types. Even by changing just theproportions within the two cross-sections, the qualities of the DPF canchange widely.

3.2 Technical Procedure

This method is perhaps the easiest to use. The user enters the rowcross-section (RCS) and the column cross-section (CCS) onto an XZ-planeand a YZ-plane, respectively. Optionally the row and column repetitionnumbers may be factored out, and the data entered may be represented bythe two-dimensional vertices of the PL curves as described previously.The x-axis of the RCS is used as a reference line, and the RCS givenshould have no undercuts. In Figure REF[DPFchart], ‘A,B,C’ show threerow cross sections and the ‘1,2,3’ show three column cross sections. For‘C’ the RCS is a sine wave (not a PL curve) and so the DPFs generatedare outside the simplest class. Note for pictorial quality the data isnot to scale and the RCS and CCS are shown to two repetitions, while theDPFs are shown to at least three repetitions in each direction.

The idea is to position each facet-row in three-space, and then attachthem along their common edge-rows. -RCS will denote the reflection ofRCS about its x-axis, and RCS and -RCS will denote the usualcorrugation-type surfaces formed by extending RCS and -RCS from theXZ-plane to three-space in the direction parallel to the y-axis.

The reference plane of these two surface is then the XY plane.Facet-rows will be taken from RCS and -RCS alternately, and positionedsequentially according to the segments of CCS. To do this label theconsecutive edges of CCS with +1 and −1 alternately. The YZ planecontaining CCS should be considered as usual to lie in the XYZ space. Oneach segment of CCS labeled +1, position a copy of RCS so that itsreference plane contains the segment, its reference plane is normal tothe YZ plane, and has not shifted in the x direction. Likewise position-RCS on the remaining edges of CCS labeled with −1.

Corresponding and parallel to the original series of edges in CCS, wenow have a series of corrugation surfaces. Each of these consecutivesurfaces should be cropped along the curve where it intersects the twoadjacent corrugation surfaces in the series. These cropped corrugationsurfaces then splice together to make the DPF.

Alternatively, RCS and CCS may be used to construct the DPF bygenerating the collection of column cross-sections of the DPF obtainedby intersecting it with each plane parallel to the YZ-plane. Todetermine the shape of these column cross-sections, consider a point(x0, z0) on the RCS. The column cross-section in the {x0} XRXR planewill be the same as the original CCS, but offset according to z0. To dothis the segments L of CCS are assigned a plus/minus value as above. Forsegment L on CCS, a new L′ is drawn parallel to L and at a distance of|z0| from L, with L′ above or below L according to the product of thesigns of z0 and L. To finish this column cross section in the {x0}×R×Rplane, the various L′ produced from the sequence of segments in CCS areextended or cropped to join their ends sequentially. FIG. 5 shows anexample.

To generate just the vertices of the DPF, the column cross-section needonly be calculated for points (X,Z) that are vertices of RCS. Thecalculations for the Method may be computed in a number of ways. Suppose(X_(i),Z_(i)) is a vertex of RCS, and L_(j−1) and L_(j) are twoconsecutive edges of CCS. The value Z_(i) represents how the vertex isoffset from its centerline. Consider the line segment L_(j−1) in the YZplane, and let L_(j−1′) be the line parallel to the segment and offsetin the YZ plane be Zi. L_(j−1′) should be offset above or belowdepending on the product of the sign of Zi and the +/−label of thesegment. Similarly define L_(j′) offset from the line segment L_(j)Calculate the point of intersection (y_(j),z_(j))=L_(j−1′)∩L_(j′). Then(X_(i),y_(j),z_(j)) is a vertex of the DPF. This is shown in the thirdframe of FIG. 5.

In summary the x coordinate of the vertices requires no calculations andthe y and z coordinates are calculated by parallel and intersecting lineformulas. The fourth frame of FIG. 5 shows the ith column of edges foundin the X_(i)×R×R plane.

4 Wave-Tessellation Method

4.1 Overview

The Wave-Tessellation Method described here produces the planartessellations on the sheet material that that specifies where the edgecreases will be on the folded sheet. Additionally the convexity of eachcrease is also specified. Of course for a generic tessellation to be thecrease patterns is impossible, and in general it is quite difficult todesign the fold crease tessellation without a system or designmethodology. The fold crease tessellation is usually given independentlyof the final structure in three-space in which case additionalinformation must be supplied. Knowing the fold crease tessellation isessential as a “parts list” showing the size and connection arrangementof the facets. The Wave Tessellation Method may also be most suited forstudying the flexural properties of a DPF under mechanical forces.

Tessellations outside DPF technology will most often have no or perhapsisolated three-dimensional positions possible. Tessellations generatedby our Wave-Tessellation Method will have infinitely manythree-dimensional positions, enabling the DPF to be produced simply, bystringing together these positions continuously, to give the Uniform DPFProcess, as described in Section 16. An explanation of the failure ofgeneric tessellations outside our Wave-Tessellation Method is also giventhere.

4.2 Constructing the Tessellation

The tessellation method is remarkably simple. Lines are drawn in theXY-plane parallel to the y axis and with periodic spacing. These willbecome the columns of edges. An example is shown in FIG. 7A. Apiecewise-linear curve with one vertex on each column-line is drawn toserve as one of the rows of edges in the tessellation (RET) as shown inFIG. 7B. This curve is only required to project onto the x axisinjectively, and should be periodic, although in practice it generallywill have slope bounded within [−3,3]. Multiple copies of the row areplaced on the plane by translating it in the y-direction. Additionallythe row may be dilated in the y-direction with either positive ornegative factors, with translates in the y-direction of the re-scaledrows placed on the plane. FIG. 7C shows scaling factors to the left ofthe rows.

For DPFs in the main class, the various copies of the row must not crossand and they should have a periodic pattern in the y-direction. Theseminimal constraints produce a tessellation that if given the foldconvexities described below, will fold into a DPF. For DPFs in thesimplest class, the rows must not intersect and a balanced convexitycondition must be met. These variations are discussed in Section 16.

The name ‘Wave Tessellation’ refers to the fact that the tessellation isconstructed by essentially drawing multiple copies of the same ‘waves’(RET) shifted and with various amplitudes.

4.3 Fold Convexity Specification

Since the edges in the tessellation are unfolded, a system of assigningfold convexities to the edges must be known in advance of the foldingprocess. For DPFs designed by the Tessellation Method above, our systemdescribed below will satisfy the internal constraints of the sheet as alinkage and yield one parameter of motion. For general tessellationsoutside our methodology, other complex systems of assigning foldconvexity may be necessary if the sheet can be folded at all.

The reader may follow the discussion on the Method for assigning thefold convexity to the edges of the tessellation along with FIG. 7D inwhich the solid lines become convex folds and the dotted lines becomeconcave folds. The edges within any single row will all have the sameconvexity. The convexity of the initial row is selected (convex in thefigure). For any two neighboring rows consider the ratio of theirscaling factors. If the ratio is positive the neighboring rows haveopposite convexities, if the ratio is negative then the neighboring rowsare given like convexities. This enables all of the rows to be given aconvexity assignment.

Next the convexity of each column edge can be calculated locally byexamining the vertex at either of its endpoints and knowing the rowconvexity at that vertex. A schematic of the possible relationshipsbetween a column edge convexity and the row edge convexity is shown inFIG. 8, with the solid and dashed lines indicating convex and concavefolds, respectively. At each vertex the row (generally horizontal in thefigure) will divide the surface into two angles that total 360 degees.The column edge that is on the larger side of the row should have thesame convexity as the row, and the column edge on the smaller sideshould have the opposite convexity. This implies the convexities of theedges in the column chains alternate each time they cross a row. As eachcolumn edge has a convexity that can be determined from either of itsendpoints, there is a redundancy to this Method that can be shown tospecify the fold convexities consistently.

4.4 Computation Formula

For computer entry data, select an initial wave W₀:R→R, seen as thegraph of a function in the XY-plane, positioned so W₀(0)=0. This is RET,and for PL waves W₀ may be specified succinctly by naming its vertices(xy). The position of wave W_(n) may then be defined inductively. Eachwave W_(n) has a pair of (y_(n),a_(n)), where y_(n) is the incrementalspacing on the y-axis, y_(n)=W_(n)(0)−W_(n−1)(0), and an is theamplitude for W_(n) relative to W₀. The sequence {(y₁,a₁), (y₂,a₂), . .. } is called CET, usually written explicitly only up to one cycle ofrepetition. Conversely from CET and RET one constructs W_(n) byWn(x):=an Wo(x)+Wn−1(0)+yn=an Wo(x)+y1+y2+ . . . yn5 Wave-Fold Method

This method is one of the most intuitive. The terminology emphasizesthat edges in a row all have the same fold convexity by viewing each rowof edges as a single “wave fold”. The user enters either a row of edgeson the tessellation (RET) or a row of edges on the folded DPF (RED), andenters the column cross-section (CCS) in the folded DPF. Depending uponwhether RET or RED is used for the row data, the method can proceedalong two different ways.

5.1 General Description of Method

Assume first that RED data is given. One of the potential applicationfor using RED is that it can be designed as the contact area where a DPFcore material meets the laminated faces, and customizing this junctionhas structural and gluing ramifications. As mentioned RED is a coplanarPL curve and can be specified by two-dimensional vertices in some AZplane. The point where RED and CCS intersect must be specified in thedata. The point will be a vertex of CCS. For convenience, the placementof RED on the XZ-plane is given with the x-axis passing through thispoint. The x axis will be used as a reference line for RED. The PL curveCCS is given by its two-dimensional vertices in the YZ plane. The Methodproceeds by considering each vertex of CCS, its perpendicular bisector,and the plane in XYZ space normal to the bisector and containing thevertex. In XYZ space each of these planes is parallel to the x-axis, andtangent to its defining vertex of CCS. On each of these planes a copy ofRED is drawn such that the reference line of RED is parallel to the xaxis and passes through the vertex of CCS that defined the plane. Thecopies of RED are all positioned on each of these planes so that their xcoordinates of RED have not changed.

The next task is to adjust the amplitudes of the copies of RED. There-scaling will occur within the plane containing the copy, with therestriction that the reference line and the x coordinates remain fixed.The remaning axis of the plane will be re-scaled linearly withpotentially a positive or negative coefficient.

The copy of RED lying in the plane tangent to the vertex of CCSspecified as the intersection vertex is not re-scaled. From here thecopies of RED are re-scaled consecutively, following the sequencing ofthe vertices of CCS that were used to construct their planes. Considertwo consecutive vertices of CCS, and the corresponding consecutivecopies of RED. We may assume the first copy of RED is scaled properly asthe Method proceeds inductively. Select a vertex of the first copy ofRED that is not on its centerline. Draw a segment from this vertex tothe corresponding vertex of the second RED. Note these correspondingvertices have the same x coordinate. Thus the second RED can bere-scaled so that this segment is parallel to the segment in the YZplane joining the considered vertices of CCS. This reduces thecomputations explicitly to simple plane manipulations and slopecalculations. Section 5.2 below carries out this Method explicitlywithout using an inductive procedure to find the scaling factor of REDand the (x,y,z) coordinates of an arbitrary vertex of the DPF.

Instead of using the row data expressed as RED, it may be moreconvenient to express it as a row of edges in the tessellation, RET.This could be valuable if one wants to control the angles and xdimensions of the individual facets in the constructed DPF.

One proceeds similarly with the planes tangent to the vertices of CCSand perpendicular to their bisectors. One then determines the foldedshape of RET at the vertex of CCS. This shape is of course RED, and sothe Method can then proceed as explained using RED. Section 5.3 explainsthe conversion process from RET to RED.

5.2 Method for a Vertex

This section demonstrates the Method explicitly, and finds the scalingfactor of RED and the (xyz) coordinates of the vertices of the DPF. Tocarry out the computations, call the ith vertex of RED (X_(i),Z_(i)) andthe jth vertex of CCS (y_(j),z_(j)); we will calculate the (i,j)thvertex of the DPF. To simplify notation, for any vector u, <u>=u/|u|.Define the edge vector leading up to (y_(j),z_(j)) byv_(j)=(y_(j)−y_(j−1),z_(j)−z_(j−1)). At each vertex (y_(j),z_(j)) ofCCS, define y_(j) to be one half the included angle, and b_(j) to be theunit bisecting vector. Thenγ_(j)=arccos(<−v _(j) >·<v _(j+1)>)/2b _(j) =v _(j+1) −v _(j)

Each copy of RED must be adjusted by an amplitude factor at each vertexof CCS. Assuming the specified intersection point of CCS and RED isgiven as (y₁,z₁) on CCS and as X=0 on RED, the amplitude factor k_(j)for RED at the vertex (y_(j),z_(j)) of CCS isk _(j)=(−1){circumflex over ( )}(j+1)cos(γ₁)/cos(γ_(j))

The (i,j) vertex of the DPF is calculated to be (x,y,z) belowx=X_(i)y=y _(j) +k _(j) Z _(i) b _(jz)z=z _(j) −k _(j) Z _(i) b _(jy)

Since there is no included angle on the first and last vertices of CCS,some convention

-   -   such as having CCS start with (y0, z0) and not calculating RED        on the ends of CCS is necessary.        5.3 Method Using RET

When the Wave-Fold Method uses RET for row data, it is easy to convertRET to RED and then apply either the recursive procedure in Section 5.1or the formulation in Section5.2. We assume again that RET and CCSintersect at (X₀,Y₀)=(0,0) on RET and (y₁,z₁) on CCS with one half theincluded angle at γ_(j) located at (y_(j),z_(j)) on CCS.

RED will consist of a sequence of segments corresponding exactly inlength to those on RET, but with their slopes adjusted. For segment(X_(i−1),Y_(i−1)), (X_(i),Y_(i)) its angle of ascent isα_(i)=arctan(Y_(i−1)−Y_(i))/X_(i−1)−X_(i))) and length l_(i)={squareroot}((X_(i−1)−X_(i)){circumflex over ( )}₂+(Y_(i−1)−Y_(i)){circumflexover ( )}₂). The corresponding segment on RED will have angle of ascentδ_(i)=arcsin(sin(α_(i))/sin(γ_(i)))

The vertex (X_(i),Z_(i)) on RED can be defined recursively as(X₀,Y₀)=(0,0)(X _(i) ,Z _(i))=(X _(i−1) +l _(i) cos(δ_(i)),Z _(i−1) +l _(i)sin(δ_(i)))

These equations enable one to convert RET to RED. Additional formula arein Section 9

6 Strip-Map Method

The Strip Map Method is a unique among the DPF procedures in the way itscolumn data (a column strip-map, CSM) and its row data (rowcross-section, RCS) emphasize certain three-dimensional properties of aDPF. While defining a strip map to be used for the column data requiressomewhat awkward data and a sub-Method of its own, there may be someadvantages of using the Strip Map Method that out-weigh the addeddifficulty: This method gives the greatest control over the silhouetteand space surrounding the DPF; It has significant flexibility forgenerating DPFs outside the main class; And it displays naturallimitations on the entry data for assuring the DPF is embedded. Beforedescribing how the strip map is applied to produce the DPF, our stripmap representation and sub-method will be explained. The terminologyStrip Map Procedure will refer to the sub-method that determines thestrip map from entry data and Strip Map Method will refer to the largerprocedure that utilizes strip maps to produce DPFs.

A strip map can be explained heuristically quite easily, as a way offolding a rectangular piece of paper to lay flush on a plane. The actualcomputations however, typically require techniques using re-iterativecompositions, piecewise defined local isometries of the plane, linearinterpolations, and/or local coordinate systems defined on each facet.Furthermore, the strip and the strip map should be represented in datastructure and methodology to interface well with designing strip mapsand application of the strip map to the Strip Map Method to design DPFsurface configurations. Additionally, the strip map should be reduced tonumerical parameters that enable convenient manipulation for both thedata entry of the strip map and the application of the strip map to DPFdesign. We have developed several procedures for representing,calculating, and designing strip maps that solves these difficulties.Before addressing the mechanism of our procedures, some formaldefinitions are given.

A local isometry f: R{circumflex over ( )}n→R{circumflex over ( )}m,n<=m is a piecewise smooth map that preserves arc length. That is if forall rectifiable γ:[a,b]→M∫|f(γ(s))′|ds=∫ _(ab)|γ(s)′|ds.

For f: G→H with G⊂R{circumflex over ( )}n and H⊂R{circumflex over ( )}mthe definition applies also. A strip is a rectangle {(y,z)|γεJ, zεK},where J and K are intervals. In practice K is usually many times shorterthen J. A strip map is a local isometry from a strip to the plane.

6.1 Strip Map Procedure 1

This section discloses our first strip map Method, the one used togenerate the images. It produces a continuous function from the strip tothe plane, by applying the folds to the strip consecutively. In FIG. 9,the fold locations on the strip are shown as segments (p_(i), q_(i)).F_(i) will denote folding the strip along segment (p_(i), q_(i)),keeping regions A_(j), j<=i fixed. The strip map will then be thecomposition F₁·F₂·F₃·F₄:strip→plane. As the points' locations changewith each successive folding, it becomes awkward to determine whetherthey should be left fixed or reflected by a map F_(i) unless someadditional data is attached to the points identifying in which regionsthey originated. For instance, use l=l(y,z) as the subscript l of theregion A_(l) containing (y,z): It is computed easily by simultaneouslinear inequalities. Let R_(i) be the reflection across line(p_(i),q_(i)). Then define the function F_(i) on labeled points (y,z,l)by ${{Fi}\left( {y,z,l} \right)}:=\begin{matrix}\left( {y,z,l} \right) & {{{if}\quad l}<=i} \\\left( {{{Ri}\left( {y,z} \right)},l} \right) & {Otherwise}\end{matrix}$

The strip map is then given using the compositionF:=F ₁ ·F ₂ · . . . ·F _(n−1)

Note that for even complicated strip maps with interior fold vertices,the Method will define the strip map by letting A₁ . . . A_(n) be a pathof regions connecting through the strip, and R_(i) being the reflectionacross the lowest edge of A_(i+1) used in the path. A rooted tree may beused instead of a path, where “<” becomes the partial ordering of thetree.

6.2 Strip Map Procedure 2

A strip map is singular on the fold creases and fold vertices. ThisMethod uses the singular set to apply a polygonal structure to the stripbefore and after folding, with the edges and vertices of thepolygonalization coinciding with the singularities or boundary, and theregions being the largest non-singular regions of the strip. These twopolygonal structures are entered as data, and formatted so that theparts of strip polygonalization before folding are easily matched withthe corresponding parts after folding in the image polygonalization.This added structure greatly simplifies the calculations within theMethod.

There are many data structures for representing polygonal structures.For the strip in FIG. 9, it would suffice to enter an array of vertices[[p₀ . . . p_(n)],[q₀ . . . q_(n)]] for the strip and the array ofvertices [[p_(o)′ . . . p_(n)′],[q₀′ . . . q_(n)′]] for the image ofstrip, where the accent indicates the point's position in the foldedstrip. This rectangular grid representation will work easily for anystrip map with vertices lying on the rectangles perimeter. In some casesthere may be triangular regions with two fold crease edges sharing thesame endpoint vertex; in this case the vertex can be listed twice sothat (p_(i),q_(i)) is still an edge for all i. For strip maps withinterior vertices as in the last example of FIG. 11 a simple array willoften be sufficient to describe the polygonalization, but again it maybe necessary to use some redundancy in the listing of the vertices. Morecomplicated data structures can be used. One representation uses a setof lists of vertices, where each list represents a polygon, by givingthe vertices in cyclic order around the polygons perimeter. In generalthe data entered need not be any more complicated then giving thevertices grouped in appropriate structure, for then with linearinequalities it is possible to determine the edges and polygons.

Once both the polygonalization of the unfolded and folded strip areentered as data with formatting to reveal their corresponding parts, thestrip map is then determined by piecing together isometries that aredefined locally on each polygon: Suppose (y,z) is a point on the striplying in polygon A_(i), and A_(i)′ is the corresponding polygon in thefolded strip. Since A_(i), and A_(i)′ are congruent, there is anisometry of the plane F_(i) that sends the corresponding parts of A_(i)to A_(i)′. In particular F_(i)(y,z) is (y′,z′). To determine F_(i),select three vertices u,v,w of A_(i) that are non-collinear. Their imageu′,v′,w′ in A_(i)′ are found without calculation, by using thecorrespondence between the compatible data structure of A_(i), andA_(i)′. Using the information that F_(i) sends u,v,w to u′,v′,w′respectively, several geometric methods are given below for determiningF_(i). Thus by representing the strip before and after folding incorresponding formats to display the vertices in a described polygonalstructure, all of the F_(i) can be determined and hence the entire stripmap. This key method has many applications to folding analysis and isdescribed briefly below and again in Section 7.

Local coordinate systems on A_(i) and A_(i)′ can be constructed so thatcorresponding points have the same parameterization. For instancebarycentric coordinates on u,v,w are used in one of our computer codes,and then the parameters are used to reproduce the correct point on thefolded strip by using the same barycentric coordinates on u′,v′,w′. Themethod in another of our computer codes uses two vertices u,v, anddetermines the unit vector n:=(v−u)/|v−u|; rotates n to find theperpendicular m; and uses this as the two basis vectors for a localorthonormal coordinate system with origin u. Likewise a localorthonormal coordinate system with origin u′ is determined. Theorientation of m′ may be supplied by either a two coloring on thepolygons of the strip or by including the corresponding vertices w onA_(i) and w′ on A_(i)′. The displacement vector s:=(y,z)−u is thenconverted using dot products in terms of the new coordinates by s₁:=n·s,an s₂:=m·s. The point (y′,z′) is calculated to be u′+sin′+s₂m′. Thislocal coordinate systems method works well, and are layed out moreexplicitly in Section 7. Symbolically we get F_(i)(y,z)=(y′,z′), whereeach polygon pair A_(i),A_(i)′ results in different F_(i).

The total strip map F is based on the individual F_(i). If (y,z) lies inA_(i), this will insure F_(i)(y,z) lies in A_(i)′, and is the correctpoint (y′,z′). Thus to determine the effect of F on a point (y,z) on thestrip, first determine which region A_(j) contains (y,z), and then applyF_(j). This is shown schematically if FIG. 0. The various F_(i) may becalculated in advance before the Method receives the input (y,z), oreach entry (y,z) may be used to find the vertices to apply todetermining the function F_(i).

An efficient variation of this procedure is to give a polygonalizationof the unfolded strip, a selection of three vertices of each polygon,and the position of the selected vertices on the folded strip. This isessentially the same as above, but the polygonalization of the foldedstrip has been reduced to selected vertex correspondence. One proceedsas above with the local coordinate systems on each polygon, with basepoints the three selected vertices. The image of a point in a polygon isthen determined using the same coordinates, but with base points thecorresponding three vertices of the folded strip. Greater detail isdescribed in Section 7.

6.3 Strip Map Procedure 3

This is our most recent Method for representing and calculating a stripmap. The idea is that the entry data can simply be the polygonalizationof the strip before folding; the Method can then calculate thepolygonalization of the strip after folding and represent the foldedstrip in compatible format with the unfolded strip; and then proceedusing isometries, local coordinate systems, or linear interpolationdefined piecewise on the individual polygons as above in Strip MapMethod 2. This reduces the entry data to just one polygonal structure,so that making design changes in the strip map is simplified.

We have several sub-procedures to construct the folded polygonalizationfrom the unfolded polygonalization.

-   -   The Strip Method 1 can be applied to the vertices of the        unfolded polygonalization, to get the vertices of the folded        polygonalization, with the edge and polygon structure induced by        correspondence.    -   Following a connection path or rooted tree out through the        adjacency graph of the polygonalization (the dual of the        polygonalization graph), the successive polygons can be attached        alternating their orientations. One way to do this uses simple        dot products.    -   The polygons may be two-colored. Polygons of one color are then        all reflected into their mirror image. The polygons are then        reassembled, reconnecting along previous shared edges. To        connect the polygons, some procedure such as a path or tree        through the adjacency graph of the polygons should be performed,        providing a sequence for connecting the polygons.    -   A rooted planar tree on the edge graph is selected. Here planar        means at each vertex the edges in the tree are presented in a        list successively (for instance counter-clockwise), with the        angle included between successive edges of the list know and        containing no edges omitted from the tree. Using this        information, there is a natural way to two-color the included        angles so that it could be extended to a two coloring of the        polygonalization. By knowing all the edge lengths, the included        angles, and the position of the root edge, one can readily        reconstruct the position of the vertices of the entire tree and        hence the polygonalization. To find the folded polygonalization,        the procedure is to reverse the sign of the included angles        assigned one of the two colors, and then reconstruct the        vertices of the tree, and so the folded polygonalization.

These sub-procedures are similar, and generally require iterative orsequential calculations to ‘build’ the folded polygonalization. Theirprogramming is simplified, by copying the data format of the foldedpolygonalization from the data format of the unfolded polygonalizationand the steps are then only required to change the vertex coordinates inthe given polygonalization of the strip. Once computed the calculationsfor the general point (yz) are computed using Strip Method 2.

6.4 Variations

Depending on implementation, these procedures can be used to calculatethe strip map on all points in the strip or for perhaps programmingefficiency the strip map's effect on just the vertices of the strip. Ifthe latter is employed one may then calculate the map's effect on theedges and polygons by interpolation as explained in Section 6.3 andSection 7, and thus construct the entire strip map if necessary forapplication. FIG. 13 shows an efficient method for calculating just thethe coordinates needed by the vertices of the DPF.

It is also possible to blend these procedures in various combinations.For instance given a polygonalization of the unfolded strip choose onebase polygon to remain stationary under the folding map. To determinethe location of a point on the strip map after folding, a path isselected from the point to the base polygon. Follow this path themethods of Strip Map Procedure 3 may be employed to determine the pointsdestination. Another combination applies the variation in the end of theStrip Map Procedure 2 Section to Strip Map Procedure 3, so that insteadof the polygonaliztion of the folded strip, only three vertices on eachpolygon are needed. One additional note, that reflections F_(i) in StripMap Method 1 are up to rigid motion of the YZ plane defined by(y,z)→(y,|z|).

6.5 Designing Strip Maps

To design a strip map, one may work experimentally with a strip ofpaper, and then compute geometrically the coordinates of the folds andvertices before folding and after folding, and then apply Strip MapMethod 2. Alternatively just the unfolded coordinates may be calculated,and apply Strip Map Method 3. To proceed without experimentation, thereis a constraint on the polygonalization of the unfolded strip: If thestrip contains internal vertices the vertex must have an even number ofadjacent edges and by adding and subtracting the angles around thevertex alternately the sum angle must equal zero.

To produce periodic folded surfaces, the strip map f:R×[0,1]→R{circumflex over ( )}₂ should be periodic, so for some minimalp>0 there is a translation v so f(x,t)+v=f(x+p,t) for all xεR, t ε[0,1]. To assure the folded structure has no interference problems, thestrip map f should also be injective in the first coordinate, that isfor all x,y εR, t ε[0,1], f(x,t)< >f(y,t), when x< >y. FIG. 11symbolically illustrates four examples that satisfy these periodic andinjective properties, by showing the edges and vertices of the stripbefore and after folding. To reduce the pictures to numerical maps, oneapplies one of the strip map procedures given above.

6.6 Designing Surfaces by the Strip Map Method

A strip map is a function F from one two-dimensional region to another,F(y,z)=(y′,z′). It may be extended to a function by G(x,y,z)=(x,y′,z′).So has no effect on the x-coordinate and is just F on the y andz-coordinates. The significance of G is that when it is applied to azero-curvature surface S it will produce a zero-curvature surface G(S).This follows as G=I×F, where I:R→R is the identity, and set products oflocal isometries are again local isometrics.

FIG. 12 show how a strip map F extends to and is applied to produce aDPF. In A) and A′) of the figure, the strip in the yz-plane is seenbefore and after applying F. Parts B) and B′) extend this to xyz-space,and the effect of on the rectangular box may be seen. In Part C), astandard corrugation surface S, having zero-curvature, is placed in thebox. The Strip Map Method applies G to S, to produce the DPF surfaceG(S).

The Strip Map Method uses F:strip→plane and a zero-curvature surface Sto produce a DPF. For DPFs in the main class, the cross section of Staken in any plane parallel to the xz-plane will give the same curve.This curve is the row cross-section (RCS) of the DPF. Note thez-coordinate of RCS composes with the z-coordinate of the strip map. Thestrip I×J used for the domain of F must have height interval Jsufficient to contain the vertical variation of RCS. Also, the plane ofRCS is perpendicular to the interval I of the strip. With theseassumptions the strip map is called a column strip map (CSM) and theMethod uses RCS and CSM as entry data.

As described above, the strip map F is extended to a mapG:Box→R{circumflex over ( )}₃. The corrugated surface S is designed tofit in the box, and then G(S) is the DPF. As it is easy to construct amap g: R{circumflex over ( )}₂→S presenting the corrugation, as anoption one may define G·g: R₂ R₃ to give the map from the unfolded planeto the DPF in three space. Whether using G or G·g, the method asdescribed so far gives the DPF as a continuous image of a local isometryfunction. This has been implemented in MAPLE computer code.

Alternatively, it is perhaps computationally more efficient and preciseto determine the effect of the strip map on the vertices of the DPF,along with the edge and polygonal structure. If needed the fullcontinuous maps G·g and G can be deduced by isometric interpolation, asexplained in Section REF[maprep]. The surfaces R₂ and S can bepolygonalized on their singular set, so that G·g: R{circumflex over( )}₂→DPF and G|S→DPF are presented by defining the vertexcorrespondence. FIG. 13 demonstrates an efficient Method for determiningthe vertices needed.

The strip map method offers an direct picture of the position of thesurface in three-space. This method is most valuable for designing DPFswith surface portions arranged in close tolerance without intersecting.To assure the DPF will embed in R{circumflex over ( )}₃, two sufficientconditions can be met. First, the RCS curve γ⊂R×[0,1] should projectR×[0,1]→R injectively. This is the usual condition. Secondly the stripmap F:R×[0,1]→R{circumflex over ( )}₂ should be injective on each lineR×{t} for each t ε[0,1] The embeddedness follows immediately from theinfectivity hypothesis on F and RCS. The surface generated will bedoubly periodic if both F and RCS have one direction of periodicity.

Additionally, we comment that the strip map method is not restricted touse rectangular strips as described here for simplicity, as long as itis a local isometry. Also a variation rotates the surface S about avertical linebefore applying G.

7 Representation for Folded Structures

In many cases it is useful to describe the function that carries pointson the unfolded sheet to their locations on the folded sheet. Such afunction can also be relate a structure on the unfolded sheet with thecorresponding structure on the folded sheet. For instance, such afunction can describe the correspondence between the components(vertices, edges, polygons) of the unfolded tessellation with thecomponents of folded surface. For fiber composites, the function may beextended to the tangent space to relate the thread direction of thesheet before folding with the thread direction after folding. Similarlyfor perforated materials, the function extends to map the perforationsin the unfolded sheet to those on the folded sheet. We have developed asimple method for representing such a function, in the case where thefolded sheet has linear facets or can be approximated by azero-curvature surface with linear facets.

The method is uniquely related to folded surfaces. Surfaces that areproduced by conventional manufacturing processes such as stamping,casting, and forging and that do not have zero-curvature can not berepresented by this method. Conversely while there are many ways torepresent a general surface, this Method is uniquely efficient in itsexploitation of the arc-length-preserving characteristic of foldingmaps, enabling the entire function to be interpolated from a sparse datapoint correspondence, using local isometries expressed through simplecoordinate systems.

The unfolded sheet has a singular set consisting of vertices and edges,as does the folded sheet. These give a polygonalization on bothsurfaces. Suppose v_(i)=(x_(i),y_(i),z_(i)) is a vertex (the z_(i) maybe omitted on the unfolded sheet). An edge can then be described bylisting its two endpoints. If v₁=(x₁,y₁,z₁) and v₂=(x₂,y₂,z₂) are itstwo endpoints, the points along the edge are described by (1-t)v₁+tv₂,where 0<=t<=1. As before z may be omitted for the unfolded sheet. Apolygon region can be described by listing its vertices cyclically,P=(v₁,v2, . . . . v_(n)). To check if on the unfolded sheet, the point(x−x _(k))(y _(k+1) −y _(k))−(y−y _(k))(x _(k+1) −x _(k))<0(x,y) lies in the polygon ((x₁,y₁),(x₂,y₂), . . . (x_(n),y_(n))) one maydetermine if

-   -   for all 1<=k<=n

The three dimensional case can be handled similarly by first rotatingthe facet to lie in the XY-plane or by using vector operations. For apoint p known to lie in a polygon region (v₁,v₂, . . . v_(n)) in two orthree dimensions, we have several methods to re-coordinatize p relativeto its polygon vertices. Three vertices u,v,w should be selected from(v₁,v₂, . . . v_(n)). In each of the examples below, new coordinates a,b (and c) are determined by solving the given equations.

-   1. Barycentric coordinates: With a+b+c=1    p=au+bv+cw-   2. Linear coordinates:    p=v+a(u−v)+b(w−v)-   3. Orthonormal coordinates:    p=ar+bt    where r,s,t is the orthonormal basis given by    r−(u−v)/|u−v|,s=(u−v)×(w−v)/(u−v)×(w−v),t−r×s

The user should choose a convenient rule for determining which verticesu,v,w of the polygon will be selected, and should choose a coordinatesystem such as one of those suggested above. Once done, the entire mapis computed separately for each polygon P by generating the coordinatesa,b above for each point pEP and then using a,b to generate p′ withexactly the same equations but calculated using u′,v′,w′ on the foldedsheet, that correspond to u, v, w.

The unique feature of this Method is that because it is describing afolding map that preserves arc-length it can exploit the fact that thedistance (intrinsic and extrinsic) relationships on each polygonexpressing the points in terms of the vertices are the same before andafter folding. This enables a coordinate system on a polygon of theunfolded sheet, with base points u, v, w, to transfer identically to acoordinate system on a polygon of the folded sheet, with base pointsu′,v′w′. One way is to use isometric interpolation along the edge andpolygons to determine the full surface. Effectively, the entire surfacemap is determined by the correspondence on the vertices, between theunfolded and folded sheet. This identical transfering of coordinatesystems is quite novel, because sheet material formed into similarstructures by state of the art methods can not be interpolated this wayand generally would require complicated non-linear relationships. FIG.14 shows a flowchart of the steps of the procedure. In the figure, xrepresents either a point p as above or a structure component on theunfolded sheet such as thread direction, perforation location, andothers.

8 Local Isometry Method

The Local Isometry Method is versatile for generating DPFs and workswith the zero-curvature properties of the DPF transparently. Itdescribes DPFs as the image of a local isometry F:R{circumflex over( )}₂→R{circumflex over ( )}₃. Points on the unfolded sheet are assignedtheir folded locations in three-space. As with the other methods, it maybe advisable to compute F just on the singular set or just on thevertices, and then apply our isometric interpolation in Section 7 if thefull map is needed.

There are many combinations of ways to construct DPFs by composing localisometries. In this section, we disclose data structures and methods forrepresenting and calculating these functions. One word of cautionhowever, in general it is difficult to determine in advance if theabstract surfaces generated by this method contain self-intersections,so some additional screening (usually on a case by case basis) mustapplied to determine the application to sheet materials. For simplicityin notation we will describe the case where the DPF is an infinitesheet, with it implied that for most applications the domain would berestricted to a finite sheet.

8.1 Background Examples

A folded sheet may be seen as the image of a local isometryF:R{circumflex over ( )}₂→R{circumflex over ( )}₃. Once produced F maybe represented using the method of Section 7. To design or create alocal isometry F:R{circumflex over ( )}₂→R{circumflex over ( )}₃ is muchmore difficult. The method in this section constructs local isometricsF:R{circumflex over ( )}₂→R{circumflex over ( )}₃ by combining severalsimple local isometrics through set products and composition. The methodis illustrated with a first example: Define Ξ:R→R by sending evenintegers to 0 and odd integers to 1 and extending linearly:Ξ(2Z)={0}, →(2Z+1)={1},and dΞ|(n,n+1)=±1, where the sign corresponds to the parity of nεZ. Ξ isa local isometry that folds the line onto the unit interval. Let I:R→Rbe the identity, then I×I xΞ: R{circumflex over ( )}₃→R{circumflex over( )}₂×[0,1] folds three-space into the unit slab.

Suppose γ: R→R{circumflex over ( )}₂ is a sine curve parameterized byarc length, and so γ×I: R{circumflex over ( )}₂→R{circumflex over ( )}₃embeds a corrugation surface. Choose a rotation ρ: R{circumflex over( )}₃→R{circumflex over ( )}₃ so that ρ·(γ×I)(R{circumflex over ( )}₂)is as in FIG. 15 a. Lastly compose with I×I×Ξ so that(I×I×Ξ)·p·(γ×I)(R₂) is as in FIG. 15 c. A priori the surface haszero-curvature; the double periodicity is verified easily using theperiodicity of Ξ and γ. In FIG. 15 b, to illustrate the idea Ξ has beenreplaced by the absolute value function ∥. The choice of Ξ, ρ and γ inthe construction are not constrained—other choices yield other DPFs. Theonly precaution is that the composition have the correct periodicity andbe injective R{circumflex over ( )}₂→R{circumflex over ( )}₃.

This example can be generated by the other methods, as seen in FIG. 4C1, where γ(R) is the RCS. If γ(R) is replaced with a piecewise linearwave, such as in FIG. 28, W1-W8, and p is any non-orthogonal rotation ofR{circumflex over ( )}₃ parallel to the x-axis, the result will be inthe simple class. The process could also be reiterated again so that(I×I×Ξ)·ρ₂·(I×I×Ξ)·ρ₁·(γ×I)(R ₂)will be a folding tessellation if the rotations ρ₁ and ρ₂ are aligned sothat the result is embedded and periodic. The key ingredient above isthat the easy one-dimensional local isometry Ξ was extended to a threedimensional local isometry I×I×Ξ. Any other one-dimensional localisometries can also be substituted into the formula.

To illustrate another composition of local isometries, select o:[0,1]×R→R{circumflex over ( )}₂ to be a strip map (with the switch ofcoordinates). Then I×σ: R×[0,1]×R→R{circumflex over ( )}₃ is a localisometry. Select γ: R→R×[0,1] to be a PL local isometry (serving as theRCS) that projects invectively. Then(I×σ)·(γ×I)(R{circumflex over ( )} ₂)is a DPF, also producible by the Strip Map Method with CSM σ (usualcoordinates) and RCS γ. But here if σ2 is another stripmap and if ρ₁ andρ₂ are rotations chosen within bounds then(I×σ ₂)·ρ₂·(I×σ)·ρ₁·(γ×I)(R{circumflex over ( )} ₂)will be a DPF.

The inventor wrote a computer program to compute these compositions oflocal isometries, and return a DPF map R{circumflex over( )}₂→R{circumflex over ( )}₃. By adjusting rotation angle the resultingsurface may be given either a periodic or quasi-periodic structure,essentially due to the harmonic patterns between the composing maps.

8.2 Computations

Computer programs composing these simple local isometries can be writtenby persons of ordinary skill in the art. To study the resultsgraphically however, computer mesh-point distances may be larger thanthe detail of F: R{circumflex over ( )}₂→R{circumflex over ( )}₃,totalling missing the structure of some of facets of the DPF. This is aconsequence of the likelihood for composing maps to interferequasi-periodically, and generate some very small facet regions.

The significance of our method is how we generate the vertices of theDPF, to maintain full detail. Making vertices explicit is nearlyessential (versus continuous function approach) since the detail in thefacet geometry with experimental compositions could be quite small(general would have limit of zero, by quasi-periodicity). To do this thesingular sets of each of the composing functions must be given a datastructure. It is difficult to use a simple array structure to define thevertices of the singular complexes because the vertex mesh becomesaskewed with each composition, so a cw-complex data structure may be theeasiest to implement. In general for two composing maps one cancalculate the intersections of singular complexes of adjacent functionsin the composition, to determine the cw-complex data structure for thecomposition. Calculating the vertices in the intersection requiressearching for which lines and hyper-planes intersect between the twomaps. This is described below, and then a simpler method is describedexploiting the fact that the composition always starts with R{circumflexover ( )}₂ on the right.

A first approach is to enter the maps defined by giving thecorrespondence between the vertices of the singular sets before andafter applying f_(i), with the edge, polygon, and polyhedron structuresgiven in terms of these vertices and the full map defined by isometricinterpolation as in Section 7. Then for f₂·f₁: R{circumflex over( )}₃→R{circumflex over ( )}₃ one calculates the intersection I of thesingular set in the domain of f₂ and the singular set in the range off₁, and use f_(1{circumflex over ( )}(−1))(I) and f₂(I) to construct thesingular sets in the domain and range respectively of f₂·f₁. Thisenables f₂·f₁ to be represented by giving the correspondence between thevertices of the singular sets before and after applying the newfunction. By repeating the procedure, longer compositions f_(n)· . . .f₁ can also be represented by the coorespondence between the singularsets of the domain and range.

Our prefered method is a variation of above, that leaves the composingmaps in their original representation, and calculate f_(n)· . . . f₂·f₁:R{circumflex over ( )}₂R{circumflex over ( )}₃ on the singular setworking from the right. Since f_(i) has a two dimensional domain thiswill simplify the calculations considerably. Many of the other f_(i) arerotations or translations and they can be left as matrixes or vectoraddition. The other three dimensional local isometries f_(i):R{circumflex over ( )}₃→R{circumflex over ( )}₃ will be products of oneor two dimensional local isometries, with a product structure on theirsingular sets. As the Method calculates f₁(R{circumflex over ( )}₂),f₂·f₁(R{circumflex over ( )}₂), f₃·f₂·f₁(R{circumflex over ( )}₂), . . .the image is a two-dimensional surface with a polygonalization. When thepolygonalization for f_(k−1)· . . . f₁(R{circumflex over ( )}₂) has beencalculated, and f_(k) is a translation or rotation, f_(k) is appliedsimply to the coordinates of the vertices in the polygonalization.Otherwise f_(k) will be similar to a stripmap and the new vertices andpolygonal structure can be calculated as in FIG. 13 with mild adaptions.In short since the local isometryF:=f _(n) · . . . f ₁ :R{circumflex over ( )} ₂ →R{circumflex over ( )}₃starts with a two-dimensional f₁ and continues to use global isometriesor isometrics with simple product structure, the calculation for thecw-structure on the singular set may be reduced nearly to thetwo-dimensional case.9 Inter-Relationships, Similarities, and Blends

In certain cases it may be advantageous to inter-blend two or more ofour methods or methodologies. For example, potentially the user couldwant to use the row data from the Wave-Fold Method (RED) because of itsdefinitive description of the row of edges on the folded DPF for perhapsbonding on another laminate, and the user may want to use the columndata from the Strip-Map Method (CSM) for its versatility or to controlthe YZ-plane silhouette of the DPF, and the user may already have thegeometrical calculations of the Two Cross-Section Method (intersectingoffset lines) on their existing software.

The similarities among the features of these five methodologies in manycases enable one to translate from the features of one procedure ormethod to the features of another. These features include the type ofinput data, the geometrical procedure, the computational procedure, andthe range of DPFs produced. For instance as explained shortly theirtypes of row data are related by trigonometric formulas, as are the CCSand CET, and each CCS can be converted to a CSM. Furthermore theirgeometrical procedures are very compatible, and allow for blending ofthe various methods. This enables easy hybrid of methods, customized tomeet the requirements of a specific application.

Two geometrical procedures were given for the Two Cross-Section Method,namely the splicing together of facet rows and an offset procedure formaking the set of column cross sections. The Wave-Fold Method gave arecursive procedure for scaling the amplitude of successive copies ofRED to get parallel column edges and a non-recursive procedure that usedtrigonometric relationships involving the half included angle of thevertices of CCS. For the main class of DPFs, the geometrical proceduresfor these two procedures are closely related in that intersecting pointsof the offset lines will always lie in the plane of the copies of REDand are equivalently determined by finding the scaling factor for RED.Moreover the location of these points may also be understood andcalculated by a reflection phenomenon, which is roughly the approach ofthe Strip-Map Method. FIG. 16 compares the geometrical procedure ofthese three methodologies. The dotted line represents a column of edgeson the folded DPF (or any calculated column cross section). Thesimilarities between the geometry of the methods can be deuced from theconguence of the figures' resulting dotted lines. Note in each case thex-coordinate of the row data is unchanged in the generated DPF, and thusit is sufficient that the figure only show the procedure in someYZ-plane.

Specific calculations, whether using intersecting offset lines,intersecting offset lines with reflection lines, applying reflections tooffset lines, calculating scaling coefficients in normal direction toCCS vertex bisector, and others are also interchangeable after applyingbasic plane geometry identities.

As for the Wave-Tessellation Method, it may be seen as the limiting casefor the Wave-Fold Method. Also after calculating the vertices of a DPFin the main class by any method, one of our computer programs willdetermine the original tessellation. To do this it calculates the edgelengths on the DPF and the facet included angles, and then reconstructsthe tessellation on the plane. Alternatively one may convert directly toRET and CET with a procedure similar to reversing the procecedures ofSection 16.

For the main class the CSM data is very similar to the CCS data. Toconvert a CSM data to CCS data, choose a z₀ in the strip, and apply thestrip map to R×z₀. The result will be a CCS capable of generating thesame DPF. To convert CCS to CSM data, apply the Two-Cross Section Methodusing CCS and a RCS consisting of a single edge [(0,z₁),(0,z₂)] where z₁and z₂ are the minimum and maximum values for a RCS that the CCS canhandle. The result will be a folded strip, that before folding wasR×[z₁,z₂]. Besides offering different conveniences on the main class,the difference between CSM and CCS lies in the way they generalizeoutside the main class.

The Composition of Local Isometries Method generalizes the Strip MapMethod. A strip map is a two dimensional local isometry, and its productwith the identity on R gives a three dimensional local isometry used forthe compositions. Additionally the Wave-Fold Method defines a planenormal to the bisector of CCS, that is used in calculations as if itwere a reflecting plane, where the process of reflecting is also a localisometry. However the Composition of Local Isometries Method allows forreiterations and other procedures.

To facilitate the conversion between the geometric procedures performedby the algorithms and methods, we offer the following explanation. If aline in the plane is reflected (envisioned as a beam of light) at someacute angle between two parallel mirrors in the plane, it will produce azigzag shape (FIG. 18 a). More complicated repetitive reflectingpatterns for a line can be produced by applying a series of mirrors(FIG. 18 b). Effectively, the CCS serves as the reflection pattern.Three of these reflection patterns are labeled “1,2,3” in FIG. 4.Reflection preserves arc length, so the resulting map induced by thereflections is a local isometry of the line. It is also possible to puta band around a reflection pattern for a line, and get a strip-map.

Likewise to reflecting a line, a plane can be reflected by adapting themirror arrangement to three-space. For example FIG. 19 a is obtained byextending the reflection scheme of FIG. 18 a. This will produce aconventional folding pattern for the plane. The resulting map induced bythe reflections is a local isometry of the plane but the image has onlyone direction of periodicity and so is not a DPF. Instead of sending aplane to reflect between repetitive mirror arrangements, one may applythe mirrors to a corrugation-type surface (FIG. 19 b).

The corrugation type surfaces can be specified by their cross section.Three of these cross-sections are labeled “A,B,C” in FIG. 4. The body ofFIG. 4 can be interpreted as showing the result of applying thereflection schemes 1, 2, and 3 (labeled ‘column cross sections’ in thefigure) to the corrugations having cross-sections A, B, and C.

To further facilitate and disclose how to mix and blend features of ourvarious procedures and methods, namely their row input data type, theircolumn input data type, their geometrical procedures, and theircalculation procedures, FIG. 17 is included showing the trigonometricrelationships of RCS, RET, RED, CCS. With this information the user cancustomize our procedures and methods by mixing and blending theirfeatures.

FIG. 17 shows the relationships between the entry data for generating asingle row-edge e on the DPF. The boldface segments under the headingsRCS, RET and RED are the edge of the corresponding row entry data usedto generate e. The additional triangles show the rise and run of thesegments on the row data plane. Note the x,y, and z axis are labeled toagree with the convention for entry data established earlier, and becomere-scaled and rotated differently under the various headings of thefigure. The CCS vertex used to generate the row-edge e is shown inboldface on the bottom piece of the figure. The perpendicular bisectorand the normal to the bisector are shown as dashed lines. The one-halfincluded angle γ is marked, and again in a congruent location. The righttriangle formed using the dotted line S is reproduced for clarity on thelower right-hand corner.

The quantities C,D,E,S,T,U,σ,τ,υ also do have geometrical significance.The angles σ,τ,υ are seen on the entry data, as the angle of inclinationof the row. The distance between the endpoints of e is E. The distancebetween the column segments, extended to parallel infinite lines,containing the endpoints of e is C. The distance between planes normalto the x-axis of the DPF and containing the endpoints of e is D. Thelengths S,T,U appearing in under the headings RCS, RET, and RED are the(usually signed) rise in the entry data as shown in the right triangles.They appear again in the CCS right triangle in the lower portion of thefigure.

With this the data types become quickly inter-formulated.

Another similarity of these five methodologies and their derivatives, isthat they all generate DPFs by combining an infinite selection of rowdata with an infinite selection of column data. The fact that our MainClass has this array structure that indexes by rows and columns is noveland versatile.

10 Extension to the Main Class

The procedures and methodology discussed so far has focused on DPFs inthe simple class. DPFs in the simple class have a tessellation withfour-sided regions meeting four to a vertex, have edges that arestraight line segments, and have an overall slab-like shape in the sensethat it that fits between two parallel planes. The main class is verysimilar to the simple class, but may have triangular regions, curvededges, and/or an overall shape that is cylindrical for instance. DPFs inthe main class can be produced by all of the procedures andmethodologies.

It is easiest to extend from the simple class to include triangularregions in terms of the Wave-Tessellation Method. Consider two adjacent“waves” in the tessellation, with differing amplitudes. (The easiestcase to see is with amplitudes of opposite sign.) By lessoning thespacing between these waves until they coincide at some points withoutcrossing, the DPF will gradually change with certain facets gettingsmaller until they disappear entirely. Facets in the same row adjacentto the disappearing facets will become triangles. Some vertices willincrease in degree.

This can be accomplished in corresponding form by each of the methods.In the Two Cross-Section Method, if the RCS is tall enough the offsetwill be large enough (see FIG. 5) so that one of the offset segmentscalculated will have zero length. In the Strip-Map Method, if the striphas triangles or internal vertices, it is possible for the DPF to havetriangles provided the RCS is positioned inside the strip to meet thevertices of exceptional degree. (Exceptional degree here means ahorizontal open half-plane through a vertex contains two edges of thevertex.) The other methods produce triangular regions similarly.

To generate DPFs with curved facets by any method, simply use anon-linear row input data. For instance in FIG. 4C, a sine wave y isused for RCS with three different examples of CCS. FIG. 15 a shows the asine wave corugation surface R×γ=S with the application of the localisometry in FIG. 15 c. FIG. 4C 1 and FIG. 15 c are essentially the sameDPF, which could also be produced by applying the top strip map in FIG.11 to this S. Other curves can be used for the row data RCS,RED,and RET,as well as pieced together linear and curved data.

For cylindrical overall shapes of the DPF, select a CCS on the YZ-planewith an overall cylindrical shape. The CCS may be a hexagon, an octagon,and other non-intersecting star shapes for example. This corresponds toa strip map of a finite strip that rejoins at the end. Arch-like DPFscan be consturcted using arch-like column data.

DPFs in the main class can be described by all of the designing methodsof Part I. There are additional extensions to the methods outside themain class. Some of the methods have extensions that do not easily applyto the others. Because of the inter-relationships between the variousmethods described in Section 9, in some cases or for some input data itwill be possible to translate to another of our methods.

There are many traditional forming technologies that can be combinedwith our technology, such as perforating or re-enforcing the sheetbefore or after forming, either generally across the sheet or incoordination with the facet positions, splicing DPFs in design ormaterial with other sheet products, and using a DPF as architecturalbase providing the substantial sheet placement, with deforming processessuper-imposed.

Part II Selected Structures and Features of DPF Technology

11 DPF Data and Pattern Type Parameterizations

This section demonstrates the use of our DPF technology to study a DPFpattern type through parameterization. This is key value when optimizinga given pattern type for a given application. Our method will beillustrated first for the DPF pattern type in FIG. 41A, called thechevron pattern in the literature. The task is to represent the chevronpattern by convenient parameters, and study the effect of parametervariations. In this example the pattern will be used as a core materialbetween two laminated faces. Since the rows of edges in the folded DPFare easily recognized as the ridges and valleys, and they describe thecontact area to be glued to the faces, in this example we optionalychose RED for our column data type. The column data type CCS is selectedbecause it imidiately gives both the pitch (the spacing betweensuccessive ridges) and the thickness of the chevron pattern. RED and CCSare shown in FIG. 25 upper and lower portions of the figure,respectively. The first segment of both waves, shown in boldface in thefigure, has been positioned with its initial end on (0,0). This way theparameters of the other endpoint of the segment entirely determine thesegment and thus through symmetry the the entire wave. The coordinatesof the other vertices are shown. Thus RED is entirely determined by(a,b), and the CCS is entirely determined by (B,C), and the chevronpattern type has been reduced to independent vertex parameters(a,b,B,C).

To generate the DPF vertices, one may subsitute values into (a,b,B,C),calculate RED and CCS (or their units), and then use the Wave-FoldMethod to generate the DPF with those parameters. Alternatively, one mayrun the un-evaluated parameters through the method to determine avariable description of the DPF vertices. To do the latter, since theplane normal to the bisectors of the vertices of CCS are all parallel tothe XY plane, the scaling factors for all the copies of RED are all 1.With this the Wave-Fold Method produces a parameterized description ofthe vertices of the folded DPF, as depicted in FIG. 26. The (x,y,z)position of the vertex in the mth row and the nth column of edges isgiven in the table in FIG. 27. Note the rows run in the x-direction andthe columns run in th y-direction, and the data was as in FIG. 25 sothat the zeroth row and zeroth column vertex is at (0,0,0). FIG. 32shows the chevron pattern under various values of (a,b,B,C) where

-   -   a=x-extension of a row edge    -   b=y-extension of a row edge        -   B=½ pitch        -   C=thickness            interpreted directly from the definitions of RED and CCS.

Of course parameters other than (a,bB,C) for the chevron family of DPFsalso can be given by using RET or RCS instead of RED, by using CET orCSM instead of CCS, and/or applying obvious identities such as usingpolar coordinates to express the entry data.

This procedure for expressing an entire family of folding tessellationsin terms of a list of parameters applies to all DPF pattern types byfollowing steps similarly to the example above. One choses the form ofthe row and column data most meaningful to the application, expressesthe row and column data as parameters, and uses the methodology of thisdisclosure to generate DPFs of the selected pattern type. FIGS. 28,29,30show many waves that can be easily reduced to independent parameters.Waves 1-9 and 13-19 21, 22 may be used for CCS, Waves 1-8, 10-12 and20-26 are intedned for RCS, or RED, and 1-3, 6-8, 10,22,23,25,26 forRET. Section 14 says many ways the waves are used to generate DPFs.

12 Mounting Plate Application and Design

The physical DPFs have many applications, including compartmentalizationof space and structural core materials. In these applications and othersthe DPF often must be attached to external mount points or laminates. Itis sometimes useful to design the DPF with convinient locations forcontacting the other object. This may provide areas for gluing, welding,riveting, or in general bonding the two together. These areas may beentire facets or partial facets of the DPF, and are called mountingplates. For instance, one application is attaching a DPF truck bed tothe truck frame. The mounting plate has many uses including attachingbrackets or other objects conviniently, and having flush contact regionsfor bonding a folded core material to laminated faces. Note thecompeting honeycomb core material only bonds on its edges to a laminatedface, as do prior folded core materials such as FIGS. 4A1 and 4C1.

To incorporate a mounting plate into a DPF parallel to the XY plane, ahorizontal segment ((a,f),(b,f)) can be included in the row data, withanother horizontal ((c,g),(d,g)) in the CCS data. This will produce ahorizontal rectangular facet of width (b-a). In the case that f=0, theheight and length of the facet will be g and d-c, respectively. Forother f the height and length are adjusted by offset corrections asshown in FIG. 16, where the quantities depend on the choice of row data(RCS, RED, RET), the included angle at (c,g) and (d,g) on CCS, and thesign of C(−1){circumflex over ( )}n, where n is the vertex (c,g) in theCCS vertex list. These quantities can be determined using the DPFMethodology of Part I. In FIG. 5, the procedure for determining offsetusing RCS row data type. FIG. 4 b 3 shows an application of this methodfor designing horizontal mounting plates to bond to parallel faces. Inthe figure the segments generating the mounting plates occurredperiodically in the row and column data, yielding an array of mountingplates.

It is also valuable to use sloped segments in the row and column data todesign non-horizontal mounting plates. To design a mounting plate in agiven location one may determine the row-cross section and the columncross section of the desired plate and spline the segments into the RCSand CCS data. Mounting plates are also useful in many multi-layerlaminations, that use folding tessellations exclusively or combinecorrugated surface, plane surfaces, or other laminates.

The design and use of mounting areas for materials formed by traditionalmethods such as stamping and casting is well established. For foldedtessellations and other complex folded structures the use of mountingplates has similar utility. Not only is this a new art for foldingtessellations, but we present designing techniques for incorporatingmounting plates into any of the folded structures in DPF technology.

13 Tie Areas

In application DPFs may be needed to meet specific physical demands,including strength and vibration absorbtion. For these needs and othersit is valuable to incorporate Tie Areas in the folded tessellationstructure. These are facet-to-facet bonding areas within a foldedstructure. Viewing a folded tessellation as an assembledge of individualfacets, forming the structure by folding a sheet is already anextrememly efficient procedure, for the connectedness of the originalsheet enherently provides the facet-to-facet connection across sharededges. Tie areas are additional bonding areas within the DPF thatprovide locations for gluing, welding, riveting or otherwise bonding thefolded tessellation to itself. The manufacturing procedure for bondingthe tie ares may occur in parallel or after the forming process, or incertain applications such as producing cylindrical DPFs it may be donein advance.

One technique for producing tie areas is to design the foldedtessellation to have edges positioned so their included angle is 0degrees, that is the edge is completely folded. In this way interiorportionss of the facets sharing the edge are in flush contact, with thisarea being ideal for bonding. The technique can be applied across rowedges and across column edges in DPFs. Tie areas can also be utilized tobond facets that do not share an edge of the tessellation. In FIG. 28waves such as W9 can be used as CCS to accomplish this.

To design tie areas to bond adjacent facets sharing a common row edge,one may use RCS or RED row data with a vertical edge. The FIG. 10 showsentire columns of facets, generated with a vertical RCS or RED edge, allsuccessively connected across tie areas. The figure also shows thecolumn under our Uniform DPF Process. A detailed description of thistype of tied column is given in Section 16.1.

The Strip-Map Method may also be used to design tie areas by using astrip map devised to fail the infectivity condition on its edges. Thiscan produce tie areas both for facets that share an edge and for thosethat are not adjacent on the tessellation. Previously folded corematerials had relied on the connectedness of the original sheet fortheir physical properties. Folded tessellation having tie areas offeradditional possibilities of bonding or fastening the foldingtessellation to itself. Advantages include stronger core materials.

14 Specific Patterns

This methodology can generate countless DPFs with a wide assortment ofapplications. FIG. 28,29,30 show many wave types, which can be readilyparameterized as explained in Section 11. For RCS and RED data W1-W8,W10-W12, W20-W26 are useful, and for CCS W1-W9, W13-W19, W21, W22 aswell as many not drawn. FIG. 4 shows the array of patterns generated bya few of these waves. Furthermore there are many variations obtained bychanging the parameters.

For RCS W4, W5, W12, W20, W21, W24 will produce DPFs with tie areas. InFIGS. 1D and 4B1 the combination RCS:W4 and CCS:W1 is used to produce acore material that becomes exceptionally strong when bonded on the tieareas. A designer may antipate analytically mixing this core materialwith the chevron pattern, by trying RCS waves such as W5, W20, W21.These examples may demonstrate a tequnique for strengthening the sidesof a core panel, with W21 showing different flexural properties fromabove and below. Using RCS:W26 and CCS:W4 or CCS:W3 produces anintersetingcushioning material with many mounting plates on one side andround flutes on the other, perhaps for easily installing a cushioningbarrier.

For some applications such as packaging materials, a product maymanufactured to a volume reduced state, shipped and stored untilpurchased by a consumer, whereupon the consumer expands the material toits usable state, and puts it to application. For RCS W1,W6,W8, and CCSW1-W8 the material may manufacure to almost a solid block, to savevolume related expenses. For cylindrical materials, CCS W14-W19 offer ainteresting results. The odd shaped polygons need to be spiralledslightly. This has spiralling has value also for even sided CCS, forassembling a long faceted tube from narrow sheet material. CombiningCCS:W119 with many RCS waves will produce a core material forinterfacing between two cyllendar faces. CCS:W13 produces anotherstructural pattern for arches or curved panels.

Part III Forming Process and Material Flow

15 Sheet Process Background

There are many machines and procedures to manufacture thethree-dimensional structures produced by our patterning technology. Someinclude casting, cutting, assembling and stamping. A significant valueof DPF structures is that their intrinsic surface geometry correspondsto the intrinsic surface geometry of a sheet. This offers the intendedpossibility of manipulating a sheet into the DPF geometry withoutsignificantly stretching the sheet throughout the forming process. Themain obstacle is because none of the folds extend clear across thesheet, all of the fold vertices impose simultaneous constraints forcingthe facet motion of the entire sheet to be interlocked. Additionally,the process should be reasonable to implement in machine design.

In origami texts, a process of forming the “chevron” pattern can befound, that folds, unfolds, back-folds, re-folds, etc a sheet of paperone move at a time until the whole chevron pattern is folded. In thisprocedure no extraneous creases are made, but many creases are foldedand refolded with opposite convexity many times. Anther option for aforming process include allowing temporary folds or bends in thematerial in the interior of the facet regions that are later removed. InSection 17 another process is disclosed, of allowing a fold to migrateor roll as it is formed until it moves into its final position and thenbecomes clearly instated. Combining these techniques gives countlesspotential processes for forming folded structures.

In general one process, described by the three-dimensional flow, can beimplemented by many machines. There are many apparatuses to move anddirect a material through a cascading network of geometry. However theprocess, the sequencing and specification of the sheet motions, isessential for assuring the relative distances measured within thematerial between any two points remain constant, to prevent anystretching of the sheet during the forming of the DPF.

We have developed three processes for producing DPFs withoutsignificantly stretching the sheet, and built machines that successfullyimplement them. To distinguish our procedures from others we have calledours the Uniform DPF Process, the Novel Creasing Process and theTwo-Phase Process. The first applies to all DPFs in our main class andto some DPFs outside the main class, but generally will not apply tofolding tessellations designed outside our methodology. The secondprocess describes a novel method of producing an individual creaselocally. This creasing process is especially useful when faced with thetask of efficiently folding complicated networks of folds. The thirdprocess solves the tessellation folding problem from a very practicalproduction point of view, and applies to many folded sheet structuresinside and outside of our methodology.

16 Uniformi DPF Process

16.1 General Information

This section offers a process that solves the ‘linkage constraints’.These constraints imply the material is not stretched during the formingprocess, that no temporary creases, bends or folds occur in the middleof facets during the forming process, and that the folds creases do notmigrate. An advantage of this type of forming process is that otherforming processes that do not satisfy the linkage constraint essentiallychange the crease tessellation during the forming process, and in manycases it is desirable to have no temporary folding. The only activitythat occurs during the forming processes under the linkage constraintsis the folding along the edges of the tessellation and the correspondingrigid motion of the facets. In our solution, the Uniform DPF Process,the fold angles along the creases of a DPF are increased gradually, andsimultaneously across the entire tessellation.

Most of the prior state of the art knowledge on folding tessellationshas been done on small paper models. In these models the individualfacets do not always remain planar, and generalizations about thefolding process are easily made erroneously, and so will not apply tolarger tessellations or the tolerances of a machine. One mistake is thatif a tessellation has a three-dimensional folded position, to expect itto have a gradual folding process that satisfies the linkageconstraints. In FIG. 20, a four-sided polygon is drawn with 16 angles athrough p labeled. The polygon may be a square, a rectangle, aparallelogram, a trapezoid, or a general quadrilateral. The angles drawnare not to scale, and the figure is intended only to represent acandidate for a degree-four region, with degree-four vertices, in aarbitrary tessellation. Nearly all (the exceptional set has co-dimensionat least 2 and may be closer to 10) of these components will not foldgradually. Unless all of the angles are selected in coordination,similar failure will also occur for triangle, pentagon, and otherpolygon components with degree four vertices. For instance if a linkagewith hinges was constructed to imitate one of these component, forcingthe linkage to fold would cause it to bind and bend the leaves of thehinge, or otherwise distort the device. For a generic componentcorrecting the failure would involve changing several of the angles inthe figure whose calculation could involve simultaneous quadraticequations involving over 20 variables. Furthermore for a tessellation tofold gradually, not only must every component fold gradually, but all ofthe components must fold gradually with interlocking equations. The DPFsin the Simple Class have all of their polygons represented by thefigure, and will fold gradually under the linkage constraints because ofthe unusual geometry of our designing algorithms.

DPFs described by row-column methodologies in this disclosure have adistinct advantage over folding tessellations generated by means outsideour technology. It has been stated that it is difficult to designfolding tessellations without a methodology, for the no-stretchzero-curvature condition imposes a very complicated set of simultaneousconditions on the vertices and edge lengths of a proposed structure.Once designed, as explained above, determining a procedure to fold atessellation can be even more difficult, even if the description of thedesired three-dimensional form is precisely known. In fact most oftenthe linkage constraints have no solution. For DPFs in the Main Classdefined by our methods, not only does our Uniform DPF Process satisfythe linkage constraints, but we have numerical procedures that give theUniform DPF Process. The technology enables one to model the materialflow inside a machine and to calculate the position of the variousapparatus or computer controlled mechanisms that guide the material.

For DPFs in our Main Class the Uniform DPF Process can be specified interms of any of our designing methods. With our Uniform DPF Process isassociated an additional parameter t, called the flex parameter. Theinput row and column data can be adapted to yield the correct surface atthe intermediary stage t of forming. This is only possible because theintermediary surfaces formedfrom DPFs in the main class by the UniformDPF Process are also DPFs in the main class. This is another uniqueutility of using our methodology.

Each of the DPFs in the main class may start as a tessellation in theplane and may be folded, by the Uniform DPF Process, to increasingextent until a there is a collision of facets within the structure.Often the collision occurs between two adjacent facets when there commonedge is folded to 180 degrees (0 degree included angle). Any earliercase of facet collision would happen between facets in the same column,and can be ruled out by checking the embedding condition in theStrip-Map Method. When the collision occurs between two adjacent facets,there will in fact be an entire column of facets with their bridgingedges folded to 180 degrees. Moreover the entire column will lie in asingle vertical plane, and the segment of RCS generating the column willbe vertical. Moreover during the Uniform DPF Process the angles ofinclination or declination of the segments of RCS start at 0 degrees andchange monotonically until one of them reaches the vertical position(assuming as before the strip map is injective in the first coordinate).For this reason we have elected to choose for our flex parameter t theparameterization that runs from 0 to 90 degrees and represents themaximum of the absolute values of the angles of inclination of thesegments of RCS. In FIG. 17 this would be the maximum of |σ| over alledges of RCS. Of course t could be parameterized or re-parameterized inmany ways, for example simply composing this parameterization on theleft or right with any smooth monotone function, but for definitenessthis t is the parameterization used here.

As only the tessellation data RET and CET remain unchanged during thefolding process, while RCS, RED, CCS, and CSM change trigonometrically,we will first show our method for specifying the Uniform DPF Process interms of RET, CET, and t. This is valuable also because in Section 4 aprocedure was given for producing tessellations that would produce theMain Class of DPFs, but no method (other then experimentally working thestructure) was given for calculating the three-dimensional form.

16.2 Uniform DPF Process from Wave-Tessellation Data

To utilize the advantage of the zero-curvature structure of DPFs, theforming process also should not induce significant in-plane stretchingof the material even temporarily. The forming process may fold, bend,and perhaps unfold, unbend the material, but should not produce in-planedistortions beyond the requirements locally in the folds related to thematerial thickness and fold radius. Of course if the DPF technology isbeing used as an architectural base to design hybrids in betweentraditional forming technology and folding technology, the no-stretchcondition during the forming process may be relaxed proportionally tothe hybrid. In either case it is desirable to have a mathematicaldescription of a folding process that precisely maintains theno-stretching condition, whether applying it to manufacture a structureproduced strictly by folding or using it for an architectural base for ahybrid process.

Several methods for calculating the position of a tessellation in threespace during the Uniform DPF Process are described in the subsectionsthat follow. Other defining data for a DPF (RCS,RED,CCS, CSM and thethree-dimensional image) all change continuously throughout the UniformDPF Process, but the tessellation and tessellation data remain constant.Thus the Wave-Tessellation Method provides information that is valuablethroughout the Uniform DPF Process, while the Two Cross Section Method,the Wave-Fold Method, the Strip Map Method, and the Composition of LocalIsometries Method only give a static DPF, unless they are additionallyparameterized and supplied with data indexed by a time or the flexparameter.

This first method converts RET and CET to RCS (or RED) and CCS in termsof the flex parameter t and then applies the Two Cross Section Method(respectively the Wave-Fold Method) to generate the three-dimensionalfolded form. With the background knowledge of the procedures andmethodologies in this disclosure, the most direct approach to generatethe folded structure for RET and CET at time t is to convert the data toRCS/RED and CCS and apply the appropriate method. As in FIG. 17, letE_(i) and τ_(i) represent the length and angle of inclination of the ithedge of RET. Likewise for the other variables in the figure under RCS,RET, and RED, the subscript will correspond to the depicted in the ithcolumn in the row. The subscript m is first determined so that the mthsegment of RET has maximal absolute slope, that is |τm|=max(|τ_(i)|),where i runs over the edges of RET. We will also use Cm, Tm, and Em. ForCET the subscript j will denote the jth vertex and augmented pair(yj,aj) as in Section 4.4, where yj is the spacing on the y-axis betweenthe the (j−1)th and jth row wave of the tessellation, and aj is theamplitude factor for the jth row wave. Once folded to parameter t, theangle yj will denote one half the included angle of the jth vertex ofCCS similarly to depicted in FIG. 17.

At t=0 RCS is collinear and horizontal with its ith segment of lengthCi, determined as the x-displacement of the ith segment of RET. At timet the x displacements and column cross section elevation of the ithsegment of RET are Di and Si respectively. Define Sgn:R→R to be thefunction that returns −1,0, or 1 according to the sign of the variable.Solve forSm=sin(t)·Cm·Sgn(Tm)Dm=cos(t)·Cm

Put k=Sm/Tm. Then at time t for the ith segment of RET:Si=kTiDi={square root}(Ci{circumflex over ( )}2−Si{circumflex over ( )}2)

Then RCS is constructed recursively by(x0,x0)=(0,0)(xi,zi)=(xi−1+Di,zi−1+Si)

The method for producing RED is very similar. Put k={squareroot}(Em{circumflex over ( )}2−Dm{circumflex over ( )}2)/Tm=Um/TM. Thenthe x and z displacement of the ith segment of RED are Di and Uirespectively:Di={square root}(Ci{circumflex over ( )}2−Si{circumflex over ( )}2)Ui=kTi

The row RED is then constructed recursively by(x0,z0)=(0,0)(xi,zi)=(xi−1+Di,zi−1+Ui)

The method for calculating CCS at flex parameter t is as follows. RecallCET was expressed as pairs (yj, aj). Put k=tan(τ0)/(a0 sin(t)). ThenLength of jth segment of CCS=yjγj=arctan(kaj)

To assemble CCS an inductive calculation may also be performed using thelengths yj And half included angles γj in the YZ plane after choosingthe position of the first edge.

Alternatively the trigonometric relationships within FIG. 17 can beapplied in other orders around the triangles of the figure to arrive atsimilar calculation for RCS, RED, or CCS. After calculating either RCSor RED and CCS, the Two Cross Section Method or Wave-Fold Method orothers generally described in this document may be applied to calculatethe DPF folded to stage t.

16.3 Uniform DPF Process with Two Cross-Section Method

This section completes the steps for taking a RCS and CCS and theresulting DPF, and according to a new flex parameter, changing them toproduce the correct new RCS and CCS and DPF.

By using our Two-Cross Section Method, we have described the process ofusing our RCS and CCS data to design a DPF. Next we describe ourtechnique for describing the Uniform DPF forming process in terms of RCSand CCS. Based on the original RCS and CCS data, this technique willproduce the new RCS and CCS data corresponding to any flex parameter0<t<=90. From this the Two Cross-Section Method may be applied togenerate the DPF at state t. Schematically,t→(RCSt,CCSt)→DPFt

From the original RCS ((x0,z0), . . . (xn,zn)) one readily converts backand forth between the incremental form ((x0,z0), (D1,S1), (D2,S2)) . . .(Dn,Sn)) where D and S are as in FIG. 17 andxk−xk−1=Dkzk−zk−1=SkOne could alternatively use incremental polar coordinates (ck,σk) as inthe figure. In either case for simplicity assume z0=0.

Select a maximally sloped edge m to give the current flex parameter,σm=max arctan(Sk/Dk)

Then, for any flex parameter value t, 0<t<=90, compute the newincremental RCS at stage t in incremental rectangular coordinates bySk′=Sk*sin(t)/sin(σm)Dk′={square root}(Sk{circumflex over ( )}2+Dk{circumflex over( )}2−Sk′{circumflex over ( )}2)

Or in polar coordinates byCk′=Ckσk′=arcsin (sin(t)/sin(σm)*sin(σk))In either procedure the quantity bt=sin(t)/sin(σm) does not depend on kand could be substituted in advance into the calculations. Once RCS isdetermined it is then converted back from incremental coordinates to thenew ((x0′,z0′), . . . (xn′,zn′)). The procedure for adapting theoriginal CCS to the new CCS at stage t is similar. Convert CCS ((y0,z0),. . . (γN,zN)) to incremental polar coordinates (Lk,θk) soLk*sin(θk)=zk+1−zkLk*cos(θk)=γk+1−yk

Thenγk=(180+θk−θk−1)/2Lk=Lkγk′=tan(γk)*sin(σm)/sin(t)

Then with θk′=2γk+θk−1−180, one converts CCS from its incremental polarform to ((y0′,z0′), . . . (yN′zN′)), using the relationship expressedabove. This procedure for calculating RCS and CCS at flex parameter tcan trivially be varied using the relationships in FIG. 17. Another verysimilar technique that we have explained converts RCS and CCS into RETand CET as explained in Section 9, applies the Uniform DPF Process withthe Wave-Tessellation Data (16.2) to get the DPF at stage t. RCS and CCScan then be read off the DPF if needed for application.

16.4 Uniform DPF Process with Other Methods

The Wave-Fold Method and Strip-Map Method can also be used to study theUniform DPF Process. The procedure is similar to Section 16.3. The rowand column data are entered in advance. The flex parameter t is entered.The row and column data are converted to correspond to the DPF at staget. The DPF at stage t is then generated from the row and column data.There are many other alternatives due to the relationships in thetriangles of FIG. 17, including converting the data to apply to Section16.2 or 16.3.

17 Novel Crease Forming Process

Surprisingly, there are still new methods for forming individual creasesin sheet material. Our novel crease forming process is useful insituations where the material or forming apparatus pose geometricalconstraints on the material flow. In particular for folding complicatednetworks of folds, our new method for forming individual creases is ofgreat value in that it allows for unusual combinations of folds to occursimultaneously.

17.1 Traditional Fold Formation

Naturally, to form a crease one may etch, or mark the crease locationand then carefully fold it. In brakes and other folding machines onepositions the sheet properly and then folds it on the creasing axis ofthe machine. For continuous process the sheet passes through rollersthat instate the crease at the desired location. In each case thefolding operation is centered near the location of the final crease.FIG. 21 schematically depicts three variations showing the location ofthe fold as it is formed. Whether the material is bent softly and thenthe radius decreased until the crease is formed or whether the radius issmall and the fold angle increased, the folding is centered near itsfinal position. In the figure each drawing represents a cross-section ofthe sheet material as the crease is formed. In the first series thecurves are polynomial, the second series the segments are filleted withprogressively smaller radii, and the third series is simply an anglebecoming more acute. Of course many other curve profiles centered at thefinal crease location are used by traditional folding processes. Thefigure both represents a cross-section of the sheet drawn through andedge crease location and through a vertex crease location.

17.2 Fold Migration

Migrating folds are almost common. In the home, a wrinkled piece ofaluminum foil may be smoothed by dragging the sheet across the edge of asharp countertop. The countertop edge produces a crease that travelsacross the sheet. In the garage, on a belt sander the curved portion ofthe belt is constantly traveling across the belt surface. Viewedrelative to the belt, the curved portion rolls across the material. Thetwo examples are similar, although the countertop edge produces a verysmall fold radius in comparison to the radius of the drum of the sander.

17.3 Novel Crease Forming Process

Our process, called the Novel Crease forming Process, applies both toedge creases and vertex creases. In common situations it first appearscumbersome, or even absurd, and this non-obviousness had previouslyinhibited its invention. However this procedure has great utility whenforming creases in constrained enviroments, or under unusualcircumstances. Our process combines the two processes above. By usualstandards our process starts by ‘mis-folding’ the sheet and then rollingthe bend or crease into final location. The material is softly curved ortemporarily creased in a position not directly centered on its finallocation. The curve or crease is then caused to migrate until it reachesits desired final location. While it is migrating, typically the foldradius and or included fold angle are tightened in preparation ofinstating the crease at its final location. In some cases the migratingcrease will leave a mark in the material, but this can be removed bysecondary operation later if necessary.

FIG. 22 shows two variations on our Novel Creasing Process. In the firstseries a soft fold migrates and tightens until it reaches the finallocation on the sheet, and the second series shows a sharp foldmigrating and tightening. In the first five cross sections of the secondseries both the fold location and the fold angle change; in the lastcross section only the fold angle was changed.

One advantage of this for manufacturing networks of folds is that itallows for the sheet geometry to be ‘roughed out’ first under much morelax constraints and with more diverse geometrical tolerancess. Inparticular, the migrating crease locations allow for the tessellation tovary during the forming process. For continuous processes, our novelcreasing process may have the creases migrating into the oncoming sheetmaterial as they are formed, and greatly reduces the complexity requiredof the forming machine.

An application and advantage of our Novel Creasing Process is shown inFIG. 23. In the geometrical environment shown, traditional foldingprocess can produce creases only near the ends of the material, whileour process has much greater flexability.

18 Two-phase Process

18.1 Utility

Our Two-Phase Process solves two problems. Forming fold tessellationswithout significantly stretching the material during forming ischallenging. The difficulty of working the material locally on thefolds, while the overall size of the material contracts in both the xand y directions simultaneously, posses obvious tooling complication andexpense. Moreover for continuous processes, since one end of the sheetis folded and the other end in not, some other method not obligated tosatisfy the linkage constraints should be employed.

18.2 Continuous Machine Implementation

In FIG. 24, a schematic of our continuous machine is shown with thematerial initially pre-gathered into corrugation at A), the DPF formingrollers at B), and the finished DPF at C). The axis of the roller isparallel to the y-axis of the DPF, and the flutes and the direction ofthe material flow are parallel to the x-axis. In this machine thematerial is taken from a roll and first pre-gathered into a sine-wavecorrugation. The contraction ratio, that is the ratio of the width (they distance) of the material projected onto the AYplane to the width ofthe material if measured when unfolded is the same for the material atA) and at C). This notion of pre-gathering the material to have the samecontraction ratio as the final DPF is essential to prevent the materialfrom having to contract inside the roller in the direction of the rolleraxis. For wide rollers this would require the material to slide in they-direction over the teeth while inside the rollers, which is almostimpossible. But contraction in the x direction within the rollers occursnaturally because the forming region on the material is advanced in thex direction, and the teeth revolve to engage in the x-direction. Theprojected image of the material more rapidly enters into the rollersthan it exits.

The strategy of this material flow process was to simplify thesimultaneous two-directional contraction of the Uniform DPF Process intoindependent x and y contractions. In the first phase, they-pre-gathering into a corrugation type material is easy to perform witha series of guides or rollers; and in the second phase, the remainingx-contraction and fold crease formation is performed easily by advancingthe material through a simple roller mechanism. The machine has beentested successfully on paper, copper, and aluminum. As the DPF isproduced without stretching the material, this is also a novel use ofrollers tofold sheet material.

In the descriptions above one may also use our Continuous MachineImplementation with the roles of x and y interchanged

18.3 Batch Machine and Process

A batch process may proceed as the continuous process above usingshorter rectangular sheets. In some cases for rectangular sheets it maybe preferable to advance the rectangles along one axis first and thenalong the other axis second. In this way both phases may induce thecontraction in the direction the material is advancing. This contraststhe continuous process above, where since the material is continuouslymoving in the x-direction, the initial y-contraction occured orthogonalto the material flow. For machine implementation, a rectangular sheetmay be passed first through a mechansism in they direction, and thenthrough another mechansim in the x direction, as shown in FIG. 31. Thefirst pass imparts a corrugation to the sheet with flutes runningparallel to the x-axis. The first mechansim in simple cases could be apair of corrugated rollers. Since the first phase only requires theoverall size of the sheet (the projected image onto the XY plane) tocontract in the y direction, and the material is advancing in the ydirection, the contraction can take place in the forming region in thecorrugated rollers. The produced corrugation is desired, however, tohave y-contraction ratio the same as the completed DPF. The profile ofthe corrugated surface produced should be selected to facilitate thesecond phase.

Next the fluted material is fed in the x-direction into the secondmechansim. Options include having sharply creased or softly bent flutes,and having the y-period of the corrugation greater than, equal to, orless than the y-period of the DPF, and the overall profile of thecorrugation. Next the fluted material is advanced in the x-directionrelative to the folding mechansim. The material may be fed into themechansim, or the mechansim may be passed across the material. Thiscould be many devices, including mating patterned rollers, designed byinterpolating the folded structures geometry into polar coordinates, asexplained in Sectionl 8.5, and computer controled articulatingmechanisms. Two features of the second phase is the no signifigantstretching of the sheet material and its conversion from a flutedcorugation type geometry to the patterned folded structure. Remarkably,this operation has been carried out successfully on paper and metalfoils.

In the descriptions above one may also use our Batch Process with theroles of x and y interchanged.

18.4 Description of Two Phase Process

The Two-Phase Process gives a procedure for producing facet foldedstructures that is valuable for production. The usual complications offolding the individual edges of a tessellation or fold network, due tothe interlocking effect of the fold vertices, resulting in thesimultaneous rotation and motion of facets, and the overall contractionof the network in both the x and y directions, had previously been theobstacle to mass-production. This new process reduces the material flowproblem to two phases, each of which may readily be carried out by avariety of mechanisms. This valuable simplifcation of the geometry ofthe material flow is possible in part through the non-obvious droppingof the linkage constraints and the non-obvious feasability of phase 2 ofthe process.

In the first phase the material is pre-gathered in the y-direction, byimparting flutes or corrugations of various profiles. The contractionratio in the y-direction should be nearly the same for the pre-gatheredmaterial and the folded structure, and should be distributed evenlyenough to facilitate phase 2. In many cases the creases or bends of theprofile will be removed in phase 2, and optionally are not required tobe centered on the vertices of the folded structure.

In phase 2 the material is advanced continuously in the x-directionrelative to the contact area of the forming mechansim of phase 2. Thismay be pairs of mating rollers, articulating guides, or other formingtools. The contact tool continuously advances in the xdirection acrossthe material, trasforming the flutes of the pregathered material intodefined networks of folds. The material moves locally in the ydirection,but the overall contraction ratio in the y-direction of the materialbefore and after phase 2 does not change signifigantly. The formingprocess does not signifigantly stretch the material.

In the descriptions above one may also use our Two-Phase Process withthe roles of x and y interchanged.

18.5 Folding Rollers

The use of rollers to form patterned structures by embossing, stamping,or crushing sheet materials is well established. The use of rollers tofold sheet material into DPFs and other folded networks is new. It issurprising that a pre-gathered sheet material, such as a sine-wavecorrugation, may be manipulated in three-space under the influence ofrollers into a folded tessellation structure. The fact that flutedmaterial will convert into faceted material under a folding operationthat does not signifigantly stretch the material inside rollers iscompletely non-obvious, and extremely valuable for mass production.

With the methodology for generating DPF vertex coordinates, DPF formingrollers can also be readily designed by interpolating the slab parallelto the XY plane containing the DPF onto a shell of a cylindar. For twosuch rollers to fully engage, it is sometimes necessary to expand thevalleys while preserving the ridges of the patterned rollers.

1. A method for providing a pattern for folded sheet structurescomprising: (a) entering row and column data and an intersection pointthereof into a computer program; (b) calculating a doubly-periodicfolded (DPF) surface pattern from the data entered in step (a); (c)outputting the DPF surface pattern calculated in step (b) for folding asheet structure according to said DPF surface pattern.
 2. The methodaccording to claim 1, wherein step (c) includes outputting the DPFsurface pattern to a folding machine.
 3. The method according to claim2, wherein the folding machine receiving the output in step (c) is acasting machine.
 4. The method according to claim 2, wherein the foldingmachine receiving the output in step (c) is a cutting machine.
 5. Themethod according to claim 4, wherein the cutting machine is a millingmachine.
 6. The method according to claim 2, wherein the folding machinereceiving the output in step (c) is a stamping machine.
 7. The methodaccording to claim 1, wherein the DPF surface pattern is output to adisplay.
 8. The method according to claim 1, wherein the DPF surfacepattern is printed.
 9. The method according to claim 2, wherein the DPFsurface pattern is output to a computer numerical control (CNC) machine.10. The method according to claim 1, wherein the row data entered instep (a) comprises a row of edges in a tessellation (RET).
 11. Themethod according to claim 1, wherein the row data entered in step (a)comprises a row cross section (RCS).
 12. The method according to claim1, wherein the row data entered in step (a) comprises a row of edges(RED) of a folded sheet.
 13. The method according to claim 10, whereinthe row of edges all have a same fold convexity and are coplanar. 14.The method according to claim 12, the row of edges all have a same foldconvexity and are coplanar.
 15. The method according to claim 1, whereinstep (a) includes at least one vertex of the row data and column data.16. The method according to claim 1, wherein the row data entered instep (a) comprises incremental vectors in polar coordinates.
 17. Themethod according to claim 1, wherein the column data entered in step (a)comprises a column of edges in a tessellation, augmented to giverelative amplitudes and spacing of successive rows of the tessellation(CET).
 18. The method according to claim 10, wherein the column dataentered in step (a) comprises a column of edges in a tessellation,augmented to give relative amplitudes and spacing of successive rows ofthe tessellation (CET).
 19. The method according to claim 12, whereinthe column data entered in step (a) comprises a column of edges in atessellation, augmented to give relative amplitudes and spacing ofsuccessive rows of the tessellation (CET).
 20. The method according toclaim 17, wherein the column of edges alternate sequentially in foldconvexity and are coplanar in a vertically oriented plane.
 21. Themethod according to claim 18, wherein the column of edges alternatesequentially in fold convexity and are coplanar in a vertically orientedplane.
 22. The method according to claim 19, wherein the column of edgesalternate sequentially in fold convexity and are coplanar in avertically oriented plane.
 23. The method according to claim 1, whereinthe column data entered in step (a) comprises a column-cross section(CCS).
 24. The method according to claim 1, wherein the columndata.entered in step (a) comprises a column strip map.
 25. The methodaccording to claim 10, wherein the column data entered in step (a)comprises a column-cross section (CCS).
 26. The method according toclaim 12, wherein the column data entered in step (a) comprises a columnstrip map.
 27. The method according to claim 1, wherein the row data isentered in step (a) for a plurality of rows.
 28. The method according toclaim 1, wherein the column data is entered in step (a) for a pluralityof columns.
 29. The method according to claim 27, wherein steps (a) and(b) are repeated for each one of the plurality of rows, before step (c)is performed only once.
 30. The method according to claim 28, whereinsteps (a) and (b) are repeated fore each one of the plurality of rows,before step (c) is performed only once.
 31. The method according toclaim 27, wherein for each one of the plurality of rows, step (a), (b)and (c) are respectively performed.
 32. The mehtod according to claim28, wherein for each one of the plurality of columns, steps (a), (b) and(c) are respectively performed.
 33. The method according to claim 27,wherein the DPF surface pattern step (b) includes at least one row offacets, said facets being connected in a row and column direction sothat said at least one row of facets is bounded on either side by a rowof edges, and the facets are connected successively across column edges.34. The method according to claim 33, wherein said at least one row offacets comprises a plurality of rows of facets, wherein a row-crosssection being an intersection of a plane with a ruled surface, each onethe rows of facets being a family of parallel line segments.
 35. Themethod according to claim 28, wherein the plurality of columns areparallel.
 36. The method according to claim 1, wherein the calculatingof the DPF surface pattern begins with a general DPF pattern which isadjusted according to the row and column data from step (a).
 37. Themethod according to claim 11, wherein the RCS is based on threedimensions comprising x, y and z dimensions.
 38. The method according toclaim 12, wherein the RED is based on three dimensions comprising x,yand z dimensions.
 39. The method according to claim 37, wherein the rowdata for the RCS is supplied on the XZ plane, and the column data issupplied on the YZ plane.
 40. The method according to claim 38, whereinthe row data for the RED is supplied on the XZ plane, and the columndata is supplied on the YZ plane.
 41. The method according to claim 10,wherein the RET is based on three dimensions comprising x,y and zdimensions.
 42. The method according to claim 41, wherein the row datafor the RET is supplied on the XY plane, and the column data on the YZplane.
 43. The method according to claim 39, wherein an axis of the Xdimension is used as a reference line. 44-50. (canceled)